# Tagged Questions

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### Exercises on nets

well I'm learning nets with Munkres, but I'd like to do more exercises than those in this book. Any web site or reference would be welcome. Thanks in advance
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### If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen.

If $X$ is a noetherian topological space, then any union connected components of $X$ is clopen. This is exercise 3.6P of Vakil. I can see that a union of connected components is closed. This is ...
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### Turning an isometric embedding into a homeomorphism

While in studying functional analysis, there is a part of a homework problem from Rudin's Functional Analysis that asks to show that the isometric embedding $\phi: X \rightarrow X^{**}$ is a ...
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### Infinite Cartesian product of countable sets is uncountable

Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots$. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$. ...
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### Fundamental group of real $3\times 3$ matrices with rank $1$

In "Manetti - Topologia" there is the following exercise: Compute the fundamental group of real $3\times 3$ matrices with rank $1$. He suggests to show that there is a covering map of degree ...
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### Completion of a metric space

I got a doubt with the next exercise. Let $(X,d)$ be a metric space. Denote $\mathcal{B}(X,\mathbb{R})$ the subset of all bounded functions from $X$ into $\mathbb{R}$. Let $a \in X$. Show that ...
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### Stone-Weierstrass Theorem exercise.

Well, this is the exercise: Let $E,F$ be two compact metric spaces and $f:E\times F \to \mathbb{R}$ a continuous function. Show that for $\varepsilon >0$, exists a finite system ...
Would just like a sanity check. I don't see the necessity of the locally path connected condition on $A$. The proof that $\tilde{A}$ is a covering space seems straightforward. We use the ...