Hello, I'm Yushi, I'm a technical lead of HAD at Mankind. Today, I will talk client-side ML design strategy on the C2C e-commerce application. This is today's agenda. First, I will talk about what is client-side ML? Where do client-side MLs use? What is the advantage of client-side ML? Then, I'll share a tool stack to develop application using client-side ML technologies. And I will discuss the key points for client-side ML production development. Finally, I'll talk about the next step of client-side ML. Client-side ML, it is often called HAI as well. Nowadays, machine learning is widely used in production, focusing around web and mobile applications. The ML features are almost used in server-side training and inference. In server-side ML, data such as photos, water, and beast is uploaded to servers from user devices. And ML models are applied to the data on servers. Whereas in client-side ML, you perform ML model inference on user devices and data isn't uploaded to servers. I will introduce several use cases on client-side ML. First, Google Assistant. In this application, by talking to your smartphone or smart speakers, you can request various things to the devices. For example, to find out how to get the airport, you can simply ask to the device, how do I get to the airport by public transport? The device shows the result. Live Text. It was introduced in iOS 15. It allows you to easily copy and paste text from photos such as taken with your iPhone without any additional operation. Background replacement on Google Meet. I guess many people are using this. This is a widely used client-side ML feature or robots world. You can replace or blur your background in an offline meeting. Google IME. This is a familiar feature for Japanese. It will suggest the appropriate words depending on the context you are typing. I will share the use case in Merakari. Merakari is a C2C e-commerce platform. Customers can list an item on Merakari, and the other customers can buy one. They released Merakari Lens Beta that uses client-side ML technology. In this feature, by just holding your smartphone on an item, you can find the category of it in Merakari. The similar items sold in Merakari and the price range of the similar items. You can see the market price very easily and check it one after another since you don't need to take a photo or input any information. The client-side ML can achieve new user experience. It can achieve more human-like interface. In Google Assistant and Smart Speakers, you can request something to the device by speech, the same as the human, and more interactive experience can be achieved. In the live text of iOS, you can get text directly from image. In Merakari Lens, you can get information by just holding a camera on the item. And it can help personalize the application. In Google IME, the suggestion will be personalized more and more by running your input without uploading the very text. The suggestion model is trained on your devices. Therefore, your privacy is protected. Recently, many tools, SDK, or libraries for client-side ML has been developed. When you create an application for Apple devices, you can use Core ML. In this framework, you can create an ML model on your Mac and convert it to the mobile-available format easily. And the model can be used on sweet code. When you create a cross-platform application, ML Kit is useful. In this document page, you can see many examples of client-side ML applications such as barcode scan, face detection, OCR, translation, and more. Many use cases are supported by this example. When you try more advanced features, you can use more low-level open-source libraries like TensorFlow Lite, TPM, or PyTorch Mobile. The development steps for any library are similar. You need to train your model on the server and compare it to a specific format for the runtime. This library is TensorFlow Lite, TPM, PyTorch Mobile, and provides both converter and runtime. In this way, you can use any models you like on the client-side. So now, you can create your own application using client-side ML easily for just demos. Is it also easy to provide applications for production? Unfortunately, there are still difficulties to introduce client-side ML into production. So, I will share the strategy for development of Merkai lands. When considering production development, there are four key points to check. Reason, size, time, validation. You should consider them in order from the top according to the phase of development. The first point is reason. First of all, you need to consider whether the client-side ML is truly required. As we discussed, the client-side ML can achieve new experiences, and it has potential to improve UX dramatically. However, developing and operating it requires additional engineering costs. For example, tuning model and library for valid devices is needed. The user and the user log is limited because the data is not uploaded and inference is performed on user devices. So, analyzing log is difficult. And the model should be converted to a special format and run on a wide span of environments. So, tests for ML models become difficult. If the ML model includes the application binary, the application update is required for the model update. To avoid it, you need special infrastructure for distributing ML models. So, if the server-side is enough for your use case, you should choose server-side. When should we use the client-side? Several situations cannot be achieved without client-side ML. These are three typical use cases that the client-side ML is inevitable. First, the application requires extremely quick and stable response time for ML. For server-side, the response time is unstable due to network connection, and it often exceeds one second. If the response time should be less than 50 milliseconds, the client-side is required. Second, when the data managed in the application is privacy sensitive, and it cannot be uploaded to the server, you have to choose client-side. The last case is the ML inference in an offline environment. Needless to say, the server-side ML can only be used in an online environment. Let's take our application, Melkare lens, as an example. In this application, there are six steps for displaying the information. Getting image from camera stream, perform object detection on the image, track the detected object, crop the object region, and search similar items on Melkare. Estimate the price from similar items, and display the result. We should perform all steps on the camera stream and display the result as smoothly as possible. At first, we thought, can all steps, except for getting image and display, run on the server-side? If it is possible, it will be the easiest way. On the client-side, just calling the API of the servers. However, it is difficult to locate the object detection part and the tracking part to the server-side. As object detection and tracking should apply to all frames of camera stream, these steps must be completed in near 30 milliseconds. And if all images in the camera stream are uploaded, there are privacy issues. So, these steps should be located on the client-side. For the steps of such similar items, it should run on server-side. This step is not needed to perform for every frame. It is enough to be called only once when a new object is detected. And as the variation of items which are sold in Melcali is unlimited, so the search index becomes huge to achieve higher search accuracy. The huge search index can be located on user devices. For the process mission step, it is only called after the search step and the computation cost is low. So, each side is okay for this step. We locate it on server-side. The second key point is size. We have to be concerned about how much size will be increased by introducing the client-side ML to introduce client-side ML. ML models and related libraries have to be stored in user devices. Therefore, the application size will be increased. It is known that increasing application size decreases the download of the application. So, you have to check if the size increase is acceptable. I listed the typical mitigation measures for this problem. You should choose lightweight models such as MobileNet. And let's try further tuning model architecture for size reduction. And model quantization is an effective measure. It probably reduced the size of the model significantly. Finally, it is better to try tuning library size as well. Maybe there is room for tuning build options. In our case, we compare several object detection models designed for mobile use, such as SSD MobileNet, V2 SSD Lite MobileNet V2 and SSD Lite MobileNet V3 small. In general, there is a trade-off between parameter size and accuracy. For example, MobileNet V3 small is extremely small, but the accuracy is low compared to MobileNet V2 series. We found the accuracy of the MobileNet V3 is not enough for our use case. So, we selected SSD Lite MobileNet V2. The default SSD Lite MobileNet V2 model influences the location of the object in the image and its class, such as dog, cat, or bird. And the classification needs a fully connected layer. It is well known that the number of parameters in a fully connected layer increases significantly with the number of classes. In our use case, we don't need the classification. So, we removed the fully connected layer for the classification. Let's check the effect of removing the fully connected layer. If the model has a fully connected layer for 600 classification, the model size will be 49 megabytes. But if we remove the fully connected layer, the size will be 12 megabytes. The parameters of the model are represented by Float32. The converter for TensorFlow Lite can transform it to Float16 or Uint8. It is called quantization. By quantization, the size will be half Float16 and quarter for Uint8. However, the accuracy of the model often degrades due to the parameters being truncated to Float16 or Uint8. In Mellify Length, we selected Float16 because Uint8 has not enough accuracy for our use case. Now, summarizing what we did in the size path, the size of the default model we selected was 70 megabytes. We reduced the size to 12 megabytes by removing the fully connected layer. Then, by quantizing to Float16, the final size is 6 megabytes. Finally, it is acceptable for the production release. Next, the next key point is time. In this section, I will discuss how to make the application smooth. In video processing, images are input to the system 30 times in one second. So, if the ML model is applied to all frames, the inference time should be less than 33 millisecond. If the model inference can't be completed less than 33 millisecond, it can't receive the next inputs and some frames will be skipped. That is the cause of lagging. In the previous section, I said that we removed the fully connected layer for size reduction. It has a big positive effect for inference time as well. If a fully connected layer for 600 classes exists, the inference time is about 60 milliseconds. By removing it, the time is reduced to 32 milliseconds. It looks to be able to complete inference in StoryMix. However, it's not good enough. There are many free and post-processes for ML inferences. So, in live implementation, all processes should be completed in 33 milliseconds, including pre-process, inference, and post-process. To resolve this problem, we introduced MediaFib. MediaFib is an open-source project developed by Google. It's a cross-platform and customizable framework for StoryMix media. They provide the following samples, space detection, hand tracking, object detection, tracking, and hair segmentation. So, we introduced this object tracking example to our application. MediaFib are used to represent the steps of inference as a pipeline. For example, getImage, pre-process, and object detection, post-process, and display result. These steps can be represented as one pipeline. The nodes in the pipeline can be processed in parallel based on their dependencies. The inference node depends on the pre-process node, and the post-process node depends on the inference node. So, while the inference node is performed for n plus 1 frame, the pre-process can be performed for n plus 2 frame. In this way, all steps, including pre-process and post-process, is not needed to complete in 33 milliseconds. MediaFib enables you to represent complex pipelines, which can be further optimized. In medical terms, the object detection and object tracking are performed in client-side. The detection receives image as an input, and the tracking receives image and detected object. Therefore, the pipeline can be defined like this. The detection depends on only the image node, and the tracking depends on the detection and image node. The object detection is the most compute-intensive and time-consuming node as the deep learning model. The object tracking we use with traditional computer vision technology with deep learning is not so much compute-intensive node as object detection. To reduce computation costs, we can optimize the graph like this. The tracking node can take its own output as input, and the detection node is called only several times in a second. In this pipeline, once an object is detected in the detection node, it continues to be tracked in the tracking node until it is updated in the detection node. With this optimization, the whole computation cost is drastically reduced while maintaining the smoothness of the UX. The last key point is validation. This is the most distinctive part of client-side ML. For the client-side ML, you may have already noticed the environment of ML inference is not fixed. It's a user device, and the inference time depends on the device's spec. For the client-side ML, you may have already noticed the environment of ML inference is not fixed. It's user devices, and the inference time depends on the device's spec. If the device is high-end, the inference time is short and birth-bad. Therefore, benchmarking and validating on real devices are needed. To resolve this problem, we have been developing a UX validation platform called Jetfire. On this platform, we can validate model inference time and storing inference performance and search accuracy. I'll show you a demo for it. Like this, you can check the model inference time on real devices. This is an example of Android devices such as Pixel 3 and Galaxy 9s. The smartphones have several computational units such as CPU and GPU. You can check the performance of the computational unit as well. For example, this is the result of CPU and GPU for each example Android devices. According to this result, the GPU is significantly high performance compared to CPU. So focus on GPU performance. The Galaxy 9s is much faster than Pixel 3, but the inference time of Pixel 3 is 2 9 milliseconds. So it's less than 30 milliseconds. The inference time looks acceptable. And then you can check the streaming inference performance on real devices. This is the tracking result of Google Pixel 3. And this is the tracking result of Google Pixel 5. On this platform, so you can validate the tracking work properly on every devices. And this is the validation for the search accuracy. This is the example items sold on Melcari. And this is the estimated prices and estimated categories. So this is the category estimation accuracy for each categories. And this is the score or price for each categories. By this platform, you can validate the search accuracy. So this platform enables you to validate that your application can provide the expected UX to a wider span of user devices. I have been talking about client side email for native mobile applications so far. From here, I will talk about the project that is currently under development. The next project is for the web. Web assembly called WASN is a cross-browser binary format that brings near native code execution speed for the web. Recently, with many major browsers supporting WASN, client side email on mobile browser is getting a lot of attention. The main merit of web is that customers can use the feature without application in installation. So we can reach customers who haven't created an account or installed the application yet. Both TensorFlow Lite and Media Vibe can be compiled to WASN binary. So we believe that we could migrate from native application to the web easily. We thought the application would be running browser just building TensorFlow Lite and Media Vibe into WASN binary. Unfortunately, it was wrong. There are two big issues. First, WASN thread is not supported on mobile browsers. In medical lens for native application, the process pipeline uses large thread. So our pipeline can run on mobile browsers. Second, web assembly SIMD is not supported on iOS browsers. SIMD is a type of parallel processing and it improves the performance of machine learning influence. The thread performance degraded drastically with SIMD. So we needed to modify the architecture of the application. This is the architecture for the native application. All processes are in the Media Vibe and TensorFlow Lite is used internally. This is the new architecture for the web. We introduced TensorFlow.js and redesigned the pipeline for the web. TensorFlow.js is an open source library. It enables high performance machine learning influence on web browsers. It supports WebGL and WASN pipeline. So it can achieve high performance influence with WebGL instead of WASN. The deep learning model influence in object detection is performed by TensorFlow.js. The object tracking is performed in Media Vibe and the algorithm is implemented. The tracking pipeline and its algorithm is able to run on one thread. Finally, we could migrate the application to the web. We replaced TensorFlow Lite with TensorFlow.js and use a part of the pipeline on Media Vibe. And the tracking algorithms had to be re-implemented. Now we achieved native-like performance on web browsers. This is the conclusion of my presentation. Recently, many tools for client-side ML are available. So you can create demos easily. But when you think about introducing client-side ML to production, there are four key points. Reason, size, time, validation. For web, there are still many limitations compared with native applications. However, the situation has greatly improved recently. So we are looking forward to the future development about WASN related technologies. Thank you for listening.


What they gon' do with me now? I'm still a troke of the town, talking to the sisters, I'm hoakin' them down, reachin' the smiles and the frowns, gang hop up, then we clarin' the pr- What's up, guys? We're in a new episode of Talk of the Town. Today, we got a special guest. Myra Dorn. You know that? So, for the- You wanna chain this custom, walk through when I'm lookin' like a hollywood, baby, womanage from the body, it's musin', this dress look good, cause I'm- The people that don't know, tell me from. I'm from Brooklyn. Period. Brooklyn. Yes, I love the Brooklyn energy. So, what was it like, I guess, growing up in Brooklyn? You know, it get gritty in the city, but I love Brooklyn. That's my home forever. Okay. What's the best part of Brooklyn? The best part? Yeah. What's the best part? I'm from Brooklyn, too, but I don't hear what you say. You put everything on the spot. Best style. Best style? Best style. Okay. The energy, the food, like- Everything. The energy, the- Come on, you know what I'm sayin'? Best style, breadwinners. Okay. Best style, breadwinners. Best style energy. Yeah. Well, what's your thing is the best part of Brooklyn? I feel how diverse we are.  Right. East, get a different vibe. Right. I feel like we diverse. We don't really need to go to the other boroughs, you know? Right! Yeah! Yeah. All of that. So, I guess, like, how did you get started into music? So, I started rapping, like, seriously, cause I'm in my management. He was like, you should rap. And I was like, okay. And then I started rapping. Just like that. Just like that. All right. So, walk us through that first studio session. Okay. So, it was a lot harder than I made it seem. I'm trying to make it like, you know, he just told me rapping. Yeah, that's not how I went. I showed him my first rap. He gave me five minutes to write it.  I was like, okay, that's real, but I'm gonna do better. So, it took, like, you know, a couple months of me writing and, you know, trying to, it got to a point where he was like, okay, I would let you record this. Cause he actually owned the studio. Yeah. And so he was like, I'll let you record this. And then it got to where I'm writing every day to, you know, build, grow, get better. And I'm recording every week. And so it's like a year-long process to where I am now. So, what's like, I guess you're writing process. You need to be inspired. You're trying to learn different bars. So, lately it's just been, I'm going to the studio. I'm here, you know, to pack the producer's sign. I like this one. Give me, you know, 10 minutes. I figure out babbles is what I call it. You know, I just like kind of mumble over the beat. I'm gonna say this. I might say that. And then I'll write. And then, yeah. Okay. So, you have like the freestyle in point? Yeah. Yeah. That's what I do.  And then, so do you have like a favorite producer right now or are you still feeling people loud? You know, I'm still feeling people loud. You know what I'm saying? They can send them in. They can send them in. Yeah. Yeah. You can send them in. You know what I'm saying?  So like Jason, shout out Jason. Jason, shout out Sean. He produced healthy. So yeah, I'm working closely, you know, some producers. Okay. So, music wasn't always the thing? Do you think your manager like really pushed the music thing for you? Nah, I always love music. I always wanted to be a rapper. I just never took myself seriously, you know what I'm saying? I grew up listening to hip hop. Like it's just, it's always been a part of me. I just never allowed myself to take it seriously until I met my management. What I needed was a push. A little push. Yeah. Okay. So who you listening to back in the day? Like who was inspiring to you? So actually when I was a kid, I got an iPod and my uncle put music on it. So he put 50 cent first album on it. He put the Wayne's first two albums on there. I had Jay-Z's Blueprint 12 in the black album. Had a bunch of requests on there. So that's what I was listening to at eight, nine years old. Okay. So do you feel like one of those people inspired you to start running? All of them. All of them. You know what I'm saying? When I rap, I try to, I try to, everything artist wise, Jay-Z. That's the goal. I take inspiration in everything I do when I come from Jay-Z. But like just somehow like how I rap in terms of like how cocky I am. How descriptive. I get a lot of that from Rick Ross because you could be driving a 19 or 89 Honda Civic. When you listening to Rick Ross, you driving a rave. You know what I'm saying? He just make you feel expensive. And I really like that.  Dude look good.  Right. So I guess so what kind of beats were you looking for? Did you want that kind of sound? You wanted to try something new? You talking about now? Like I guess when you was getting started.   Like I don't make music like that.  That's not me. We're going to show our koochie Wali, but we're going to be like, you know what I'm saying? Gangsta while doing it.  Like send me them little baby beats. You know what I'm saying? Okay. That's what I wanted. So you're still on that now? Or you think your energy changed? No, no, no, no. I'm still on that. It's just, I think when I first started rapping, I wasn't nearly as talented as I am now. You know what I'm saying? I'm growing every day, learning every day. So of course I'm more open, but I'm very choosy production wise. I'm very like, you know what I'm saying? I get that from Jay-Z. I got to be, you know, it got to be real crazy. Okay. So how much I would just say involved all you are. So you say you into the production, are you into like the music video set? No, I'm involved in everything. I record next to the engineer. Okay. I'm watching everything you're doing. Everything. Everything you're doing. I'm telling you now. Now it's just over. Do this, do that. I'm doing, I'm watching everything. I'm so extremely involved in my creative process because it is so important to me. I want everything to come out exactly how I hear it in my head. Exactly how I see it in my mind. Like it has to be perfect. Okay. And then same thing with the music videos too? Absolutely. Okay. So you already had Tuffy like planned out when you was going to shoot it or like, how did that go? Nah. So Tuffy having mad fast actually. So when I dropped the snippet, Tuffy wasn't even mastered.  Why not? Then I went crazy. I was like, okay, they really like the song. So let me, you know, go back to the studio, do what I got to do and like master it. And then we're going to shoot the video. So this all happened within a week. Okay. Like I dropped the first snippet. It's going crazy. I said, ooh, okay. Okay. Next day is going crazy. Yes. Okay. Let's master this. Let's, you know, maybe find the director to do the video. Find the director. And yeah. So this is all of within a week's worth of time. Okay.  Because we ain't seen the next video yet. I was you. You waiting? We waiting. You know what I'm saying? So it's in the works? Yeah. I'm always working my mind. I'm working every day. You know what I'm saying? I'm, I'm going front. This new show I'm sending on the channel. Tuffy's life. Okay. I ain't going front. Y'all thought y'all thought Tuffy was something.  So they, you have a day. A day. A day. Like when we go expect it. So before the year. Come on girl. Why are you doing all that? Come on. Before the year. Listen, right now we focused on tell me focused on, you know, it's been a month. It's been, I think a month today. Exactly. Since I dropped it. So, you know, let me, let me get this out. Yeah. I've been out like three minutes. Yeah. Please. Some artists be like, oh, I'm hot. I got to get it all out right now. Yeah. But, you know, I'm doing what makes sense. Making the right moves. I just realized we're 20. Low key. Oh yeah. So bad things. Yes. So, um, yeah. Yeah. You know how artists be like, yeah, I got to drop every week or every month. Yeah. No, no, no, no. You want to let the song breathe a little bit. Yeah, I want to let the song breathe. And you know what? I'm not in no rush to put out the next song because I know it's going to be fire. Okay.  Now I have to do something really crazy.  You know what I'm saying?  You know what I'm saying? No. So I'm chilling. I'm not too eager. Okay. Okay. So, I know. So when did it like, I guess start picking up for you for real? For real. When the snippet drop. When the snippet drop. From when I drop the snippet, it's been straight up. Okay. Gangsta. All right. So what is that? So I know you felt like going up, like who was some cosons, I guess, early on. Some cosons? You mean like people who like. Yeah. It's commenting, sharing it. Because I think you all in the shave room. Yeah. Vlogs, man. All right. They loving the kids. They loving the kids. Um, you know, a lot of people was early. I say dream though. Okay.  Flo Milly. Mona Leo. Dope. Some real dope bitches.  Okay. Okay. Okay. I want a party with Courtney. Um, but yeah, a lot of dope people. Yeah. I've seen that you had opened up with Flo. Flo? Two days back to back.  So good. Yeah. So how was that? How did you get the call? How did you get the news? So it was crazy. Um, Flo just DM me. She was like, hey girl, I love you. Um, I'm performing in New York in two days. So I want you to perform. I was like, okay. Okay. Say less. Say less, baby. I'm pulling up. You know what I'm saying? When you come to the tent, you gotta check up with the done. Shame me. Okay. I was like, oh my God, what if they don't like me? What if they don't know who I am? Like, oh my God. And I'm saying, and we are urban. So, you know, the second level of urban. I'm saying the second level of urban watching Leo perform. And I feel somebody stimming at me.  So I turn like, hello. And the guy goes, oh my God. It's my at the dawn. And the whole Lord of Ghosts Creed. And the whole Lord of Ghosts Creed. And the whole Lord of Ghosts Creed. And the whole Lord of Ghosts Creed. And the whole Lord of Ghosts Creed.  I was like, oh my God, no stop. She's performing. Watch her. Don't pay attention to me. I like ran away. I was like, don't do that. Like watch her please. And it was just so awesome from. So from that point on I was like, ah, I'm chilling. I ain't even nervous. Like niggas know who I am. Niggas know my box. Shame me. Yeah. And I went on stage. And I killed and I was eating it. And I was telling it. And I was like, okay. And then she was like, come back tomorrow. Okay, all right. So was you like prepping or you always been stage ready or like, what's the purpose? Listen, listen, the proof is in the pudding I was born to do this, you know what I'm saying? Like, I was born to do this, but I was on my Beyoncé ship. I mean it's me having me running and rapping my song at the same time. You know, they don't do that nowadays. They don't, let me tell you something.  My respect goes out to you because I was like, I was like, wait, time out. Like 30 seconds in.  I got asthma.  Like, I can't, but that definitely put me for that. Like, in terms of my breath control, I be right with so come out there. I was like, thank God. Thank y'all. Thank y'all for noticing because bitch, it took a few of my asthma pumps to get there. Okay. I'm gonna pass out. I was on, I was looking pale and red and ugly. I'm stressed. But you did it. I did it and I killed that. Because you, you've seen that. You've seen it. That's it. I've seen all of that. Okay. So, um, so are you excited for next one? Yes. Yes. Yes. It was so amazing. I was so happy for that. Like everybody loved it. And I was like, I will live on the stage if you want me to. It was so sweet. Okay. Okay. Okay. So that was my first time performing ever. Ever. Ever. Oh my God. Yeah. So flow is really so. Yes.  She could get anything she want.  I'll be like, I like, give me a second. Okay. I love that you like super like for the girls. I'm so, I feel like I'm, I'm, I'm just, I'm gonna be everybody bestie. I was like, hey bestie, hey bestie. I'm gonna be everybody bestie. I'm gonna be everybody friend. I love everybody. I feel like that's a good energy to have especially in the industry right now. You feel like the industry is like competitive right now. You know, I haven't been in the industry long enough to comment too much. No, no, no. I feel like right now. I've been outside. Yeah, I've been outside, but you know what? I'm not gonna say that because everybody has been super embraced and like, they've just been embracing me and just being super kind and super sweet to me. I think everybody that I've come in contact with has been super genuine and super sweet.  I feel like if you just nice, then it's no problem, right? Right. Right. It shouldn't be no problem. It shouldn't be no problem. Yes. So, um, so they were saying that they want to remix. Who said that? People and I was talking about it on the radar and people was in the comments like, yo, we really want to remix. Oh, they said that? You ain't see? Don't act like you don't see that. Are we gonna get a remix? Maybe. I don't know what she's talking about. Girl, I see nobody say that. I mean, we'll comment again if you want to see everything. We want a remix.   Now we never know what happens. You know what I'm saying? Just stay tuned. Stay tuned. We'll get me in my bag. I'm gonna explain this. Babe, I be in my bag. All the time? I'm in my selfie all day long. All day long. Yeah. You know what he's talking to? Okay. So, I'm gonna give you a scenario.  Okay. All right. So, you aren't, well, this one, this one is just name it. Okay. So, you're on your way to a sneaky link named five things you put in your bag. Five things I put in my bag? Yeah. I put a toothbrush in the air. I do that in the morning. I'm seeing, I'm seeing some sexy lingerie. That's what I'm seeing. I'm seeing some, oh, a bonnet and a scarf. Okay. Because? That goes to one bonnet scarf. Yeah. Bonnet scarf. Maybe a little vibrate. A secret. And I'm seeing, maybe I'm seeing a phone charger. Can't let my phone down. All right. Next one. Okay. So, you want to put your new book, but you see your ex and his new book. Okay.  You want to keep it cute? I'm telling you why, if I'm out with a new nigga, why would I care about the old one? Okay. And what he doing? And if he with a new bitch, she can't be better than that. And I thought I'm comparing. You know what I'm saying? But yeah, if I see your ex, I'm looking past him. I'm not looking at him. That's a ball. Okay. Wake that down. Wake that down. Okay. That was cute. All right. So, you see your friend on social media and her stories.  Are you in your bag or you want to keep it cute? You know, I would be in my bag, but keep it cute. Like I'm gonna, I'm gonna be salty. I'm gonna be like, oh why? All right.   So you're gonna do what you want. I feel like, you know what I'm saying?  You're gonna do what you want. But watch this. Watch how I move a quarter knee. Okay. It's gonna be my turn. And you will come. Okay. Red ship. So, it's a let me crush slides in your bm's. You want to keep it cute or you in your bag? I'll keep it cute. I have it already. So you're gonna close that. You're gonna lock that down. Are you chilling? He knows what's up. We're gonna get that. We're gonna get that. We're gonna get that. But the next one. Okay. One of your ex-boos from my attack, let me say. Hit you up right now. We're currently like, no, we're going up right now. Yeah.  Happened already. Okay. Ignored. You mad.  Yeah.   That's funny. All right. So, the last one. Okay. Are you calling me? But he don't answer but you see him posting on social media.  I'm in my bag. Cause, what's you doing? What happened? Why are you being weird to me?   But you know what? I got to be true because I don't pay your phone bill. Ignore me. But when it's my turn. Okay. Yeah. Checkmate. Checkmate. Checkmate. Okay. All right. And then, yeah, that was the last one. No, one more, one more. All right, your friend. Texas you, I went before your birthday dinner. She can't call. Bring it back or are you true? I'm gonna ask her what's wrong. I'm telling you. Cause why you can't come friend? That's my best friend. So obviously got to be something wrong, right? Yeah. But are you not or are you just gonna chill? I'm telling you. Or it gotta be an emergency. It gotta be an emergency. Okay. It gotta be an emergency. If she just not coming cause she don't want to come. Okay. When I say, you know how we be exaggerating sometimes. But you better pull whatever one. Yeah. You better pull whatever one. You better come in the PJs. In the PJs. Shammy. Don't play with me. All right. So we're gonna run back a little bit. You say, so that we crush it in the end. So you single right now? I got a couple of boyfriends. Couple of boyfriends. Yeah. All right. So what's your type? My type? I like them brown skin. I like them here. So I'm not like I'm intelligent. I like them when they stay out to mix.  Because I don't like rappers. Okay. I don't like rappers. So your group is definitely not a rapper? No. He's definitely not a rapper. Okay. He's definitely not a rapper. But he intelligent. He smooth. He sexy. He's so well spoken. Well spoken. Well spoken. And he gonna do it to me. Oh, okay. All right. Shammy. Okay. So she will have it at this time. And him is gonna spoil me. And him does this thing. Oh, I like, you know what? I'm gonna stop talking. I'm gonna stop talking. So do you feel like because you have good energy, like people think like you flirting with them? Yes. I can see that. Yes. All the time. I'm just nice. Like, and I'm everybody's friend. Like I was saying earlier. I want to be everybody's best friend. I want everybody's like, you know. So I'm always like, hey, huh, huh, huh. And people be thinking I was flirting. But you know what else?  I be like touchy feely. You know, I have to dial that back. And I have to realize not everybody's your friend. You know what I'm saying?  Cause you know what I'm saying? Bitch look good.   Are you beautiful? I guess all I'm beautiful every day. Yeah. I mean.  So like, I just don't be picking up on it. Like when they just be flirting.  Like, it's really obvious. So I be like, oh, you want me? Don't do that. Yeah. Yeah, like chill out. Like I'm one of the, I'm one of the members. You know what I'm saying? Okay, all right. So you're cool. Chillin. All right. So how I guess, how has your relationship been? I guess the family stuff. I know you're probably more busy than usual now. Yeah. But how you been adjusting to all the new things? You know, I'm, I'm, I'm, I'm keeping at home. I'm still me. You know what I'm saying?  She's still going to tell me to wash them dishes. You know what I'm saying? That's always humbling. You know what I'm saying? So my family don't really treat me different. Like my close family, my fuckers I talk to every day. Like there is no different with them. But like people, like cousins, like, because you my cousin on my daddy's side, four times removed. What are you doing?  Bitch, I ain't seen one like eight years.   I don't even, I ain't see you at grandma's house. That's, that's, if I ain't see you at grandma's house, bitch, we not cousins. Okay. Like, yeah, if we wasn't spending the night at grandma's house together waking up eating mad grits and eggs, we not cousins. We not cousins. Yeah. Okay. So, um, but I mean like, so I swear, okay, so family's good. So what's your like background? What's my background? Yeah, like Caribbean. Oh, yeah, yeah, I'm, I'm Puerto Rican. Puerto Rican? Yeah, I'm Puerto Rican and black. Okay. Yeah. So we're going to get like, I guess, like a Spanish colony. So you're not Spanish? Yeah, I'm Spanish. Oh, I did not even know she was Spanish. I definitely didn't know she was Spanish. Yeah. Are you into like other Spanish artists? Yes, I love Spanish artists. I like Spanish music, but you know what? I like old Spanish music. Like, but um, I'm into like Dembao and Bachata, but like I really like, um, like Mexican. Yeah. Yeah. But Baroni, Michael Jackson.  I see that. Hell yeah. I see that. So, so you're gonna, okay, maybe get some music like that. Yeah. For sure. Yeah. Okay. So what's the plan? What's your, what's some of your more career plans? I just, I'm a superstar. I'm not a female rapper. I'm a rapper. And then after I'm a rapper, I'm a superstar. You know what I'm saying? So it's, it's, it's bigger than brown paper bag money. You know what I'm saying? Then you money. Yeah.  For sure. Everybody know that one. Okay. Everybody know that. Don't make me, no, because I really, really like your energy and I really like kind of like what you brought to the city. It was really like a bad bitch, cute type of, oh, I don't do that. Yes. I was like, I was just like a breath of fresh air because you know, like everybody's like doing drill right now. Yeah.  I'm not a drill rapper. I'm not a drill rapper. I'm just a rapper. But shout out to all my young drill girlies killing it. Shout out young Devon.  Shout out Naya G. Shout out Kenzo. Shout out all of them like bitches is killing it. But I'm, I'm not a drill rapper. It get bigger than that. Okay. And then you went to I'm Scarlet's. We love love. Yeah. I'm like the other night. Oh, I know exactly why Drake be rapping about them strippers the way he do. Cause this one stripper, I was like, girl, I want to take you home just so we can be besties. I don't even like girls.  Like, yeah.  It was so funny.  I was just trying to hand them the stack. Like here you go. You don't even have to twerk. I was like, here you go. And I was like, oh my God, girl, your hair's eating it. Oh my God. I love your outfit. Where'd you get it? Like I was just like, I love you guys. We can all be friends. And it was just like, and I was like, wait, let's just talk. I was like, what are you doing after this? Yes. Darlis was lit. Them strippers. Oh, yeah. So that was the first time I had to strip clothes? No, not my first time at the strip club. So you went to the strip club in other states? Yeah. Because they say Atlanta would be super, super, super tight. You know what? I went to the strip clubs in Miami. I went to a lot of them. Like even though I was out, everybody would be saying it's popping.  Oh, really? You're the first to say.  Nah.  Word. Word. That's the fact. And they only stay on the floor. Everybody was talking about Tissies. Tissies was wack. I ain't going to front. Tissies was wack. Damn, maybe I shouldn't say that. We ain't mean that. We ain't mean that. Nah, I meant it.  I ain't going to front. I wasn't starting. It's like, I don't want to go home. I don't want to go home. Yes. So I'm so like, I feel like you're already bubbly. But when you drink, are you extra bubbly? What's the vibe when you drink that? When I drink, I like to get straight to the point drunk. So I just be taking shots, shots. And then I'll have like, I probably do like four shots. And then I'll do mixed drinks until I feel it. And then I'm like, take me home to my man. You know what I'm saying? Like take me home to my man. I don't even want to be here no more. You know what I'm saying? So I'll be telling when I'm drunk though. When I'm drunk, my whole goal is to not look drunk. So I'll be in my head. Like I'll be drunk as hell. Like act normal, act normal. Don't don't let them don't let them catch you slipping. I just be like, I know I'll be looking like a lunatic.  Now I always hold it together anytime. First of all, I don't, I don't really drink like that when I'm out in public. When I drink to be drunk, I'm in a crib or like when my friend, you know what I'm saying? So I know I'm good.  I was like this. You just had that energy. They'd throw you a drink. Yeah. And motherfuckers probably thought I was lit, but I was just like. I'm paying lunch. I'm alert. Yeah.  So I'll be like, you know. And then you went in the studio, seen the studio five of y'all. That was my brother's surreal. Thought we locking in songs. Got a way to see. Like what's going on? You know what I'm saying?   Shout out 5e for even letting me hijack his song like that. Like it was, yeah, it's fine.  That's my brother for real. Like that's going to be crazy. I got so much stuff coming. I got so much stuff coming. That stuff I ain't even posted yet. Okay. I mean like the Yossi song. Yeah. Tune in. Tune, you make sure. So like so, but a lot of times when you post like the videos, you always put Tuff in the background. Is that like a marketing strategy? You just want people to remember that that's you? Like what's the tactic behind that? Could be both. A lot of times when I when I put, like if I'm in the studio, I put Tuff in the background. You know why? Because you know what's playing. Yeah. Yeah, I can't hear that yet. And then also I want you to know that anybody I'm in the studio where you best believe I'm in a face like, Hey, I have this song and it's called Tuffy. Do you want to listen to it? Maybe. Maybe. I don't know if you like it. You can let me know. Of course. Listen, anybody having another studio, what they know exactly who I was. Of course. Shemi. Shemi. Name ring bells. What? What? Yes. So, um, so you connected with people out of town. Have you traveled yet? You been traveling? Yeah. Okay. So you linked with people out of town? Yeah. You know what you're saying? Yeah. No. But that's the stuff we didn't see. That's the stuff we didn't see. Yeah. I like to go OT. Okay. Yeah. What are you going to carry? You're laying those things out. I love it up there. I love it up there. Man, I'm man, I'm man. I'm up here and I'm up here and I'm back here and I'm up there. Yeah. Okay. But do you feel like you're always like living in New York? Absolutely. Okay. That's how I'm doing it. Why everybody saying that?  Now you're not.  Where you gonna go? North Carolina?  North Carolina ain't nothing.   You love New York. I love New York. I don't think I ever leave. Okay. You just drink it. I can't go here. Slow down, girl. That's you. So, um... I'm sorry, Greg. I'm getting late. You feel like you're getting late? No, I don't want you to feel like I'm getting late. I'm just... No, no, no, no. No, no, no. You go, you go, you go. Oh, God. We telling. So, um... So, I guess, have labels been calling? Are you still independent? Like, what's going on? I'm still independent. So they can still call you? So, I mean, it's all that. Maybe. Maybe. If you follow me on Instagram, you know exactly who to contact. But have you heard anything you like? You're still shopping around. You're gonna chill out a little bit. What do you mean, like shopping?  No, I just... I was a joke. Ah-ha! But I've been... But doing that, too. You ain't changed? But I'm... Do you feel like you want to like, to drop more records? Wait, what? Yeah. Yeah, yeah, yeah. Like, I started earlier. Like, I'm just doing what makes sense right now. You know what I'm saying? Of course, I'm trying to push healthy. It's only been a month. That's still an infant right now. But yeah, you can expect more. Like, a lot more. Bigger than music. So... Okay. Do I'm playing or doing like, any like business investments or any other plans like that? All right. A lot more. Bigger than music. Okay. Okay. You don't see that. Okay. Ha-ha!


Okay, now let's go to the withdrawal list and then let's go have a look at the tournament because it's Indian Wells and It's a big big tournament on the men and the women's side We had through the WTA yesterday and some interesting names there that we're missing and also some interesting matchups But the men's draws out as well So let's start with the withdrawals from the men because we do have a couple of updates and maybe some Changes that we thought maybe gonna be withdrawals, but they're not so here are they confirmed withdrawals We have Roger Federer. He is confirmed not to be playing We all already knew that Garen pulled out Nishikori also pulled out songers out Duckworth Vanola's Ema Dominic team they're the big names that are out. We can add to that list a couple more people including We have Magga and also Mulcan have pulled out of this one. All right, let's start at the top of the draw now And we got dental mevadev number one seed the top 32 seeds get a buy in the first round first time We're seeing dental mevadev the world number one playing as the world number one officially So we're gonna see him play he'll play either the winner of a qualifier or Popper and who are going out in the first round Then we've got Cranovich versus Lyovitch Winner of that takes on Gail Monfeast the 26th seed in the second round Then we've got Alcarez the number 19 seed taking on Taking on either McDonald or fooch of it. She was going at it in the first round Then we've got another qualifier in this part of the draw taking on Gaston winner of that takes on the number 15 seed Batista a goose so not a terrible draw for Mehdi I mean Alcarez not fun to play and he doesn't want to play RBA because he's a nightmare Monfeast in the third round I'm not sure their head-to-head, but I Think it's a manageable draw for Mehdi I think he would like these slower conditions second part of the top Section or the second section of the draw that in the top quarter of the draw with the defending champion Cam Norrie The number 12 seed by in the first round He play either Martin is or a qualifier in that second round match Then we've got Fnini versus and do her winner takes on the finalist from last year number 18 seed But still actually so straight away We could get a replay of the Indian Wells final of last year in the third round. That's gonna be fun 25th seed Hashinov He'll take on either Brooks be on by Anna who are going at it in the first round Then we've got Serendolo versus sock the winner that will play city passing the second round number five seed So Steph versus sock. That's what we want. That's what we want as the unofficial We are the sock train. Are we with the we are the I drive the train and you guys are all passengers on that sock train Let's see that happen, but I'm excited to see the Norrie Basile It's really rematch. Let's see if that actually comes to Comes to play as well in that third round. We could get a replay. All right talking about big names Let's go to the third section. We've got Rafa the number four seed. He is Waiting for the winner between quarter and a qualifier. We're going out in the first round So it could be an interesting second round match there for Rafa Corrie versus Quavos winner of that plays number 27 seed Evans in the second round Then we've got a pelka the number 17 seed He'll play either gear on a moussetti who were going at it in the first round Then we've got Chorich the return of Chorich. He takes on Fakina winner of that match takes on Shapa Val of the 13 seed now straight away. You can see Shapo Rafa Fourth round that could be happening That's a match that I think we all want to see because we saw it at the at the Australian open and I wouldn't say they've got beef Shapo, he's not the biggest fan of Rafa at the best of times He did have some comments about Rafa So we do want to see those guys play again and see if Shapo does make any more comments about the time The Rafa takes between points. Let's go to the next part of the draw We have Sinner the number 10 seed he takes on the winner of Kwan and Jero who are going out in that first round Then we've got a couple of Frenchmen going at it in Bonzi versus Rindinik winner of that match takes on the 21st seed Senegal in the second round then we got the number 32 seed Delbonus. He awaits the winner of Kirios He's the wild card in this one taking on Baez. We've got a qualifier taking on Cressy winner of that takes on the number 8 seed Casper Rude so I mean look the big name there is Keeg's Nick Kirios He just went and watched LeBron and the Lakers play. He also watched Steph Curry play in that match He's got a good draw Kirios, you know Baez is a beatable player on on a hard court for Keegs And Delbonus is more of a clay quarter too So we could be seeing Rude versus Kirios and we all know they got beef not to mention Sinner at the top I would say Sinner's got a good draw. I think Sinner should get to the get the fourth round and play Maybe Kirios or Rude all right now We're going to the bottom half of the draw and we have Baratini number six seed He'll take on either Umber or a qualifier in the second round after getting by in the first round Then we've got Bagnus taking on Laksunin winner of that match takes on the number 30 seed Harris in the second round Then with the 24 seed Chilich he awaits the winner of a qualifier versus Ketjmenovic Winner then we've got another qualifier in this part of the draw taking on Vandezanshub winner of that takes on the number nine seed FAA We haven't seen Felix since those two you know back-to-back finals a few weeks ago winning his first title So I I'm excited to see FAA He's got a tricky draw You know Chilich isn't easy and Vandezanshub is tough and Baratini fourth round potential But I I think FAA out of this part of the draw is probably the most confident next section of the draw now And it is the Sasha section Let's go to the top of the draw though of this part of the draw We've got the 16 seed Grinja Buster He's playing a qualifier in the second round no matter what because we've got two qualies going at it Then we've got mattress act for taking on Betonet winner of that match takes on the number 20 seed Fritz in the second round Then we've got the number 29 seed Diminor. He waits the winner of a Vashka taking on Strouf Battle of the big hitters there Then we've got Dmitriyov versus Paul that is a fun little first-round match the winner of that match will take on Sasha's Vera the number three seed in the second round It's very of his draw is good I mean I know Dmitriyov and Paul are gonna be a tough opponent no matter who it is But he can beat Diminor he can beat Grinja Buster He could I mean Fritz is a bit of a nightmare because he lost the Fritz here last year, but he can beat Fritz More than more times than not right? I would say every you know seven out of ten times He'd probably win all right So the bottom quarter of the draw now the top half of the bottom quarter if that makes sense Seven seed Rubler by in the first round he takes on the winner of Coppa or Pair We're going at it in the first round Then we've got Manorino versus Nakashima winner of that plays against the number 28 seed the big foe Tiafo who I think might be able to do some damage here. We watch out for Tiafo Then you got the number 22 seed Carrotsev he awaits the winner of Johnson versus Altmaier Then you got Gasgate taking on Ott winner of that match takes on the number 11 seed who be her catch who look Gotta be honest hasn't had the greatest start to the year didn't do great in Australia I think he did he did well in Doha or Dubai a couple weeks ago But hasn't hasn't really been the top ten guy that he was last year. So we'll see maybe can turn around here He's also the man the Miami open champion So he has points to defend over the next couple weeks, but you can see there rubler versus her caps That's a possible matchup in the fourth round number 14 seed Shapo He gets by the first round he take on either a qualifier or Russavari in the second round Then we got Gricksport versus Quarry winner of that plays John Isner number 23 seed in the second round Then we got Bublick the 31st seed he takes on either Murray who's a wild card He's taken on a quali in the first round Murray. So that's a good start there for my for our for Muzza Then we've got Thompson versus Gophane winner of that match takes on the number two seed Novak Djokovic. Yes. Noley is on the list. He's on the keys in the tournament Now we don't know if he's allowed in the country, but if he gets into America He's playing Indian Wells all the speculation about will he play wonnie play Is he allowed to play will Indian Wells let him play? It's all out the window because he's playing if America says yes So that's what we're waiting for waiting for the government for the CDC of America to say yes or no to no They I can't wrap is the easy pick I guess. I mean I'll pick Djokovic if he's playing If Djokovic is playing I'll pick him because I think no way I said that in Dubai First tournament back. He looked rusty. I Would say if Djokovic does play he is I'm gonna say he's the favorite


And now for the face on the tap room floor Jack Benny Well Well, that was certainly a nice introduction You know Wilson as an announcer you're about as useful as an umbrella to a flounder That's all right Jack. I can take it But I know some pretty swell affairs that have been run off about master of ceremonies. Oh, you do a yeah Say what is a master of ceremonies anyway? Well a master of ceremonies is a fellow that's handsome Dressed as well knows a lot of good jokes and is generally cute You're not at all conceited are you know, I'm not I'm not a bitty get this to go in fact I'm much better looking than I think I am Anyway Wilson, I think there should be less announcers on programs and more masters of ceremonies Yes, there should be more masters in the cold cold ground That's all right. That's I can take it But am I burning up folks right now you could fry eggs in my forehead?  Okay Well the glasses are off. What are you going to do about it? Nothing. I just want to see if you had blue eyes Well, listen Benny just a word for your own protection, you know, I was one paperweight champion of California remember that well I can fight too You ever hear of carnera? Yeah, well who didn't? Big smart Alec and Another thing one more word out of you and I'll take that nose of yours and put it around back of your head Just as I thought He thought I was gonna answer Peter said one more word. I'd have cleaned up the studio with them. What's that? I said the studio needs cleaning up.  Hello, Mary.  Oh nothing, Mary I had an argument with Don Wilson and I had to tell him where he got off. That's all Nothing I just have to take off his glasses and he wouldn't so I pasted him Went down like a log No, Wilson Then he got up and made a pass at me, but when I got through with him his face looked like a New England boiled dinner Then he gets sore Mary and pulls a knife on me Yeah The good thing I held my temper after all he has a family and has work to do You know Jack here comes Wilson back again. Don't mention anything. I said Mary's might embarrass, but you know after all we're on the same program See you later.  Where's your crutches crutches? Yeah See you don't look a bit like a new England boiled dinner. What in the world are you talking about? Well, weren't you in an accident or something? Certainly not Mary. He didn't jack Benny. Uh, you know, I don't understand Can you still do your work? Of course, Mary listen the general tire minimizes road and car track bump It's blowout proof runs on 40% less air and can be bought at your own general tire dealer. What else did Jack do to you?  Oh Pardon me, I was just talking to some friends of mine I'm sorry for the way that I acted when the program first started tonight. Oh, that's all right. Yeah, you know those arguments help to kill time It's all good materials And why there are more oh, I don't know I was just thinking of mother's day and Frank Parker just Loved by neighbor and all the whole thing just got me. Yep. It does be real Adam's apple up Doesn't it see I wish I didn't live so far from home You know Jack, I'd like to bring you out to the house with me for the weekend But folks all know you listen to your broadcast and oh, I'm sure that you'll have a nice rest and some home-cooked food Well, that sounds good done ready to live in the block just a short walk from the last subway stop. Well, I'm Sorry down if I haven't anything with me, you know no caving outfit or anything Oh, we've got everything there for you. Come on. I won't take no for an answer. Well, all right after all I like home cooking and I certainly need a rest When do you want to go right now? Come on. All right, okay with me That's the place something Jack's coming home with me for the weekend. Okay. Shall we take the subway down? Yes, and then it's just a few minutes walk from there all the fresh air will do it good. Okay, kid Oh Yeah, it was a long subway ride down, huh? Thought we'd never get here. Well, Jack, it won't be long now and this walk will do it good How far is it all about six minutes as the crow flies? Well, that's all right. If you happen to be a crow, but my feet hurt, you know He is quiet out this way, isn't it? Oh, yes, but you can sleep like a pop She is It's certainly dark out this way.   I say the weekend is starting nice One squawk out of you and your goose is cooked. He thinks I'm Joe Petter. Why Why is Jack quiet quiet? He means business. Come on you big fella hand over your dough I'm sorry, but I haven't anything with me. How about you? I'm sorry, but oh, yeah All right here. What's it on your finger a war?  You mean this ring if it's cheap ring I got on a merry-go-round. Can I keep it now? I'll take the ring. You can keep the merry-go-round Well, that's fair.  Hey, Jack, you should have left your dough at the studio with bastard the fine time to tell me Oh, there's a house over there. See the light shining in the window. See, that's fine. Let's run. I still got my watch, you know Here's the house. Follow me across the lawn. It's certainly a czar Oh, gee whiz. Oh, you drift over the lawn board, Jack Thanks Well, here we are now. I'll ring the bell mother of sure be surprised Hello, ma John. Well, well, this is a surprise. How are you my son? Oh, hi, do you're looking? Well, ma come in come in. Who's that with you? Oh, this is Jack Benny. Who Benny? What's the last name? Just call me Rudy Here mother, I brought you a hundred dollars for a present. I wonder where it was when we were held up Thank you, son and ma. I brought you a package too


So, welcome to this lecture. So, in the previous lecture we define continuous maps and we had entered the lecture by this lemma which I had left as an exercise. So, let us just prove this lemma perhaps some of you are able to do it. So, let x and y be topological spaces let f from x to y be a map of sets and we want to check that f is continuous. So, let b be a basis for the topology on y then if f inverse v is open for every v in v then f is continuous. So, this lemma gives us a convenient criterion to check whether a function is continuous in terms of checking only for basic open sets. So, prove. So, let u contain in y be an open set then for each x in u there exists a basic open set v sub x such that x belongs to v sub x and v sub x is contained in u. Therefore, this shows that we can write u as a union over all the x's these v sub x. So, then it is a straightforward checking set theory that f inverse u is equal to union f inverse of v sub x each of these is open as this v sub x belongs to b and by hypothesis assumption is that f inverse v sub x is open and as arbitrary union of open sets is open this implies f inverse u is open and since this happens for every open set u thus f is continuous. So, next we want to show that the space for continuous functions which are real valued or complex valued maps are is a nice has some nice algebraic structures. So, to do that we are going to begin with the following theorem. So, we have the following two. So, before that consider with the standard topology and there are two natural maps which we can define. So, then the following maps. So, the first one is the addition map this is given by a of x comma y x plus y and the second is the multiplication map this given by m of x comma y is the multiplication. So, both these are continuous. So, we will use the above lemma to prove this theorem. So, proof by the above lemma. So, it suffices to show a inverse of b epsilon z. So, let us say z is a real number and epsilon is positive or the standard topology on R has this as basic open sets and we have to show that these the inverse images of these are open. So, in order to do that that x comma y be in order to and recall that we had defined this as epsilon x comma y to be those x prime comma y prime in R 2. So, is that x prime minus x is less than epsilon and y prime minus y is less than epsilon. So, note that if x prime comma y prime is in this set then the absolute value of minus x plus y is less than equal to x minus x prime plus y minus y prime which is less than equal to epsilon strictly less than epsilon. So, this implies that the absolute value of a x prime comma y prime minus a x comma y is strictly less than 2 epsilon for x prime comma y prime in S epsilon x comma y. So, this shows that S epsilon by 2 of x comma y is contained in a inverse of b epsilon x plus y. So, let us check why this happens to see this we need to show that if x prime comma y prime is over here then a of x prime comma y prime belongs to b epsilon x plus y, but we have checked that if x prime comma y prime is here then a of x prime comma y prime minus a of x comma y is strictly less than 2 times epsilon by 2 which is epsilon, but this a of x comma y is precisely x plus y. So, this implies that a x prime comma y prime belongs to b epsilon x plus y. So, we have simply rephrased the above computation in this statement.  So, let now using this we will prove that in the inverse image of all basic open sets are open b in R be a basic open subset. So, suppose x comma y is in a inverse of b epsilon z this implies that x plus y belongs to b epsilon z. So, this b epsilon z is b of z sorry and this is the epsilon ball around z and x plus y is somewhere over here. So, this implies we can find epsilon prime positive such that this b epsilon prime of x plus y is completely contained in b epsilon z right and from the above computation the above computation it follows that s epsilon prime by 2 is contained in a inverse of b epsilon x plus y which is contained in a inverse of b epsilon z. I should write s epsilon by 2 of x comma y epsilon prime by 2 of x comma y right. So, what this shows is that so thus for every x comma y in a inverse b epsilon z there exists some epsilon prime positive such that s epsilon prime by 2 of x comma y is contained in a inverse of b epsilon. So, therefore, by the definition of the standard to polygenar 2 thus inverse of b epsilon z is open in R. So, this shows that the addition map. Similarly, let us show that the multiplication map is continuous. So, we prove the second part now. Once again we start with a small computation let us start with a delta which is positive and less than 1. So, and suppose that prime comma y prime belongs to s delta x y. So, this is in R 2 ok. So, then the absolute value of x prime y prime minus x y this is equal to x prime y prime minus x y prime plus x y prime minus x y. This is less than equal to x prime minus x into y prime plus minus y which is strictly less than x prime minus x into mod y plus delta. This because since y prime minus y is less than delta. So, this implies that mod y is mod y prime is less than mod y plus delta plus mod x. So, this is strictly less than right now this is also less than delta this quantity over here. So, we use that and this quantity is less than delta. So, plus is equal to delta into mod y plus mod x plus delta which is strictly less than delta into mod y plus mod x plus 1 ok. So, this computation shows that. So, we can this refreeze this computation as when we apply m on s delta x comma y this is going to be contained in b of delta into mod y plus mod x plus 1 of x y. So, once again let us check why this shows this this because if x prime comma y prime is in this set s delta x comma y then m of x prime comma y prime which is equal to x prime y prime. We have just now check that over here is belongs to of x y. So, sorry there is a type over here. So, now let us so let ok. So, we fix any epsilon between 0 and 1 and define delta to be equal to epsilon by mod x plus mod y plus 1. So, then clearly delta also lies between 0 and 1. So, as a result what happens is so this implies that m of s epsilon by mod x plus mod y plus 1 x comma y is contained in b of epsilon x y or which is same as saying that s of epsilon by mod x plus mod y plus 1 of x y is contained in m inverse of b epsilon x prime. So, now with this computation in mind. So, suppose x comma y belongs to b m inverse of b epsilon z where z belongs to r and epsilon is positive. Then there is an epsilon prime which is between 0 and 1. So, this is r z and this is epsilon and x y is somewhere over here. So, we can always choose epsilon prime and further assume that epsilon prime lies between 0 and 1 such that this ball of really epsilon prime around x y is contained in this ball of really epsilon around z. So, from the above we conclude that s epsilon prime by mod x plus mod y plus 1 of x y is contained in m inverse of b epsilon prime x y which is contained in m inverse of b epsilon z right. So, once again given so therefore, given any point x y. So, given any point x y in m inverse b epsilon z we have found this positive quantity such that this basic open set around x y is completely contained m inverse of b epsilon z. This implies that m inverse of b epsilon z is open. So, this shows that the multiplication map is also continuous. So, this completes. So, next we prove a similar theorem, but about r minus 0. So, let r star be the set we move 0 from the real line. So, with the subspace topology. So, and consider there is a very nice map on this set. So, consider the map there is a natural map given by f of x is equal to 1 upon x and the claim is this map is continuous. So, let us prove this claim once again. So, x ok. So, let us begin with an x in r star. So, one checks easily if epsilon is strictly less than mod x then b this basic open set is completely contained in r star. That is obvious because this is let us say this is 0. So, this is r star we have to move 0 and if we take any x and we choose an epsilon which is less than mod x then this neighborhood this interval around x will be completely contained in r star and the thing is. So, it is easy to check the collection b epsilon x where x belongs to r r star epsilon is strictly less than mod x is a basis for the subspace topology. So, therefore, we will check that. So, thus it suffices to check f inverse of b epsilon x is open when epsilon is strictly less than mod x. So, in this situation so, what do we have? We have we have removed 0 and let us say r x is somewhere over here. So, we take this neighborhood epsilon x minus epsilon and x plus epsilon then one easily checks that this set f inverse of this interval x minus epsilon comma x plus epsilon is precisely equal to the interval 1 upon x plus epsilon which is open in r star in the subspace topology. So, this proves that f is continuous. So, we will end this lecture here.


Hi and welcome to the session, let's work out the following question. The question says, using matrix method solve the following equations, 2x minus 3y is equal to 1, x plus 3z is equal to 11 and x plus 2y plus z is equal to 7. Let's start with the solution to this question. First of all, let's write the given equations in matrix form we will have. 2 minus 3, 0, 1, 0, 3, 1, 2, 1, multiplied by matrix x, y, z is equal to the matrix 117. Now let a into x be equal to b, this implies that x is equal to a inverse b if mod a is not equal to 0, we call this 1. Now mod a is equal to the matrix 2 minus 3, 0, 1, 0, 3, 1, 2, 1, this is equal to 2 into 0 minus 6 plus 3 into 1 minus 3, this is equal to minus 12 minus 6 and this is equal to minus 18 which is not equal to 0 as determinant a, this is also determinant a, so as determinant a is not equal to 0, therefore a inverse will exist. This implies that a inverse is equal to 1 upon determinant a into adjoint a, now adjoint a is equal to the matrix C11, C21, C31, C12, C22, C32, C13, C23, C33 where C11 is this into this minus this into this, similarly C21 is minus of this into this minus this into this, C31 is this into this minus this into this and so on for other elements also. So solving this we get the matrix minus 6, 3 minus 9, 2, 2 minus 6, 2 minus 7, 3. Now since a inverse is 1 upon determinant a into adjoint of a, so a inverse is equal to 1 upon minus 18 into matrix minus 6, 3 minus 9, 2, 2 minus 6, 2 minus 7, 3. Now putting the value of a inverse in x equals to a inverse b we get the matrix x, y, z is equal to minus 1 by 18 into the matrix minus 6, 3 minus 9, 2, 2 minus 6, 2 minus 7, 3 into the matrix b that is 117. This implies that the matrix x, y, z is equal to minus 1 upon 18 into the matrix minus 6 plus 33 minus 63. In the second row we have 2 plus 22 minus 42 in the third row we have 2 minus 77 plus 21. This simplifies to minus 1 by 18 into the matrix minus 36 minus 18 minus 54 and this further simplifies to the matrix 2, 1, 3. So our answer to this question is x is equal to 2, y is equal to 1 and z is equal to 3. I hope that you understood the solution and enjoyed the session. Have a good day.