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By definition topological spaces are clopen and then their boundaries are empty, but for example, is said that the boundary of the closed unit interval is it's two endpoints a so on. whath is the meanining of "boundary" in this context?

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There are two notions of boundary in mathematics. One is the boundary of a subset of a topological space, and it is a relative notion: it takes as input two objects, a topological space and a subset of it. As a subset of $\mathbb{R}$, the boundary of $[0, 1]$ is its endpoints. (However, as a subset of $\mathbb{R}^2$, for example, the boundary of $[0, 1]$ is $[0, 1]$. Again, relative.)

There is also the boundary of a manifold with boundary, which takes as input one object, a manifold with boundary. As a manifold with boundary, the boundary of $[0, 1]$ is also its endpoints.

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oh thats clarifier thanks –  Fenrir Jul 11 '12 at 20:06

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