# Tagged Questions

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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### Open and bounded set with compact boundary

Why does an open and bounded set in an infinite dimensional space have to be the emty set, if it has a compact boundary? And the space has a norm, by the way. Cheers, Richard
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### Elementary question about topology and metric spaces

Let $(X, \rho )$ be a metric space. Denote $$\mathscr{B} = \{ B(\epsilon,x) : x \in X, \epsilon>0 \}$$ Let $B_1,B_2 \in \mathscr{B}$ . Let $x \in X$ be arbitrary with $x \in B_1 \cap B_2$. ...
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### Compact subspace of $\mathbb{R}$ with lower limit topology must be countable.

Any compact subset of $\mathbb{R}_{l}$ must be a countable set. Consider the open cover $\{[n,n+1): n \in \Bbb Z\}$ of $\Bbb R$ which has no subcover. So $\Bbb R$ is not compact with respect to ...
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### If $X\neq\varnothing$ and $\tau=\{\varnothing, X\}$, then any subset of $X$ is compact.

If $X\neq\varnothing$ and $\tau=\{\varnothing, X\}$, then any subset of $X$ is compact. Disproof by counterxample? Not true. Let $X = \mathbb{R}$ with the usual topology and $A = (-\infty,0)$. ...
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### Is Every Set Open in the Subspace Topology on the Cantor Set?

Im working in the real line with the usual topology. For the cantor set subspace of R, let T represent the the subspace topology on the cantor set induced by the usual topology. I'm trying to show ...
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### Show that ${\mathscr C}(\{1,..,n\},R)$ and $R^n$ have the same open sets

Question: Let X be the set $\{1,2,...,n\}$ equipped with the discrete metric ($\delta(x,y)=0$ if $x=y$, $\delta(x,y)=1$ if $x\neq y$). Then ${\mathscr C}(X, R)$ and $R^n$, where $R$ is the real ...
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### Problem of a compact space

Let $X$ be a compact $T_2$ space.Assume that the vector space of real-valued continuous functions on $X$ is finite dimensional.Show that $X$ is finite. Spent nearly 3 hours on this problem.Cant ...
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### Continuous function that has limit at infinity is uniformly continuous (another viewpoint)

I know how to prove that, given a continuous $f:[0,\infty) \rightarrow \mathbb{R}$ such that $\displaystyle \lim _ {x \rightarrow \infty} f(x)=L$, then $f$ is uniformly continuous (by means of taking ...
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### $[0,5]$ is not compact with the Sorgenfrey topology

Show that in the Sorgenfrey topology $[0,5]$ is not compact. Justify your answer. Here is my shot at an answer. Could anyone please knock it down/improve it/help with the correct answer? ...
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### In $\mathbb R$ with the usual topology, prove that the set of rationals is not compact.

In $\mathbb R$ with the usual topology, prove the set of rationals is not compact. Here is my attempt at a proof by contradiction. If $\mathbb Q$ is compact, then $\forall \{G_{\alpha}\}_{\alpha}$ ...
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### equivalent metrics and uniform equivalent metrics

Let (X,d) be the Euclidean metric on the real number, and define σ(x,y)=min{1,d(x,y)} if if x, y are rationals or x, y are irrationals, and σ(x,y)=1 otherwise. I would like to study if these metric ...
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### Interior of a set?

I'm trying to think if their is any topology for which this is false: If G is an open set, then G = interior(G) Can anybody think of anything? I'm pretty sure it's straight forward.
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### Book on “topology ” for starters [duplicate]

This semester I have a course on topology. I'd like you to recommend me some books.
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### Proving the set of all real matrix $3\times2$ with rank $2$ is an open subset of $\mathbb{R}^{3\times2}$

I have to prove that the set of all real matrix $3\times2$ with rank $2$, M, is an open subset of $\mathbb{R}^{3\times2}$. I do not know how to do it, but trying to prove that $\mathbb{R}^{3\times2}-M$...