# 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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### How to prove that there are only two kinds of 1-dim manifolds without boundary

I just know a conclusion that all 1-dim manifolds without boundary is homomorphism to $S^1$ or $\mathbb{R}$ , but I don't know how to prove it . Why is so ?
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### A property of some Hausdorff topological spaces

Let $X$ be a Hausdorff topological space such that any closed subset of $X$ with empty interior is finite. I want to show that $X$ has only a finite number of non-isolated points. Any suggestion or ...
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### Continuous functions in the product topology on $\Bbb{R}^{\Bbb{N}}$

I'm trying to prove the following statement: Let $(X, T )$ be a topological space, and let $f : X \rightarrow \Bbb{R^{\Bbb{N}}}$ be a function, where $\Bbb{R^{\Bbb{N}}}$ has the product topology. Let ...
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### Universal Abelian Covering Space of genus two surface

Let M be a surface M, i am concerned with Abelian covers. These are the covering spaces for which the deck group is Abelian. The largest such cover corresponds to the commutator subgroup of the ...
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### Twist of irreducibility in compactifications

I am looking for a connected metric space $X$ that is (1) irreducible between two of its point $p$ and $q$ (meaning no proper closed connected subset of $X$ contains $p$ and $q$), such that (2) $X$ ...
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### Lower limit topology with completely normal

I am trying to prove that lower limit topology is completely normal. I know the it is normal. I attempted to consider this as cases let $X$ completely normal and $Y$ subset of $X$ If $Y$ countable ...
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### When proximal continuity and (topological) continuity are the same?

Under which conditions proximal continuity of $f$ (having $X\mathrel{\delta_1}Y \Rightarrow f[X]\mathrel{\delta_2}f[Y]$ for every sets $X$, $Y$ on the first proximity) from a proximity $\delta_1$ to a ...
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### Baire Category theorem and open vs. closed nowhere dense sets

In Folland's book, part (b) of the Baire Category theorem states that $X$ is not a countable union of nowhere dense sets. where $X$ is a complete metric space. It doesn't say whether those sets ...
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### Example for hollow sets whose complement is not dense in $\mathbb{R}$.

A set is hollow if it has empty interior. A set is no where dense (closure is hollow) if and only if i̶t̶s̶ ̶c̶o̶m̶p̶l̶e̶m̶e̶n̶t̶ ̶i̶s̶ ̶d̶e̶n̶s̶e̶ .̶ its complement contains a dense open set. ...
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### Proof for “Given any basis of a topological space, you can always find a subset of that basis which itself is a basis, and of minimum possible size.”

The titular statement is used in the explanation of this answer from several years back. I ran across it while puzzling my way through this text, which I bought while I was still in high school and ...
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### How to Axiomize the Notion of “Continuous Space”?

EDIT (to clear up controversy and misunderstandings caused by my poor wording): Historically, Riesz's efforts to try and make rigorous a notion of a "continuous space" (as opposed to "discrete ones") ...
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### Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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### Question about arc-connected property in a continuum

Suppose $X$ is metric, compact, connected, and $p\in X$. An arc is a copy of $[0,1]$. Is it possible that every two points in $X\setminus \{p\}$ can be joined by an arc, but there is no arc in $X$ ...
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### Is collection of all functions I-convergent to a point form a ring?

$S$ be a set. $I$ is an ideal of $S.$ $X$ is a topological space. A function $$f: S\rightarrow X$$ is said to be $I$-convergent to a point $x\in X$ if f^{-1}(U)=\{ s\in S; f(s)\in U\}\in \mathscr F(...
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### A partition of unity of a topological space

I have troubles in a little part of the following proposition. Let $(X,\tau)$ be a topological space and $\Im=\left\{U_{\alpha}\right\}_{\alpha \in I}$ an open cover of $X$. If $\Im$ has a locally ...