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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Verification: Closed Set Expands to Fill Space, but Contains No Open Ball $B_\epsilon(0) $?

I have the proof that $C$ closed, convex, symmetric in Banach space $X$ and $\cup_{n \in N \setminus 0} n.C= X $ then $B_\epsilon(0) \subset C$ for some $\epsilon > 0$. I also have the proof for $...
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1answer
411 views

Deck transformation acting properly discontinuously assumed covering space is path-connected

Let $p:E\rightarrow X$ be a covering space, $x\in X$ fixed point, $E$ path-connected and $\Delta(p)$ – Deck transformation group of $p$, that is $\Delta(p) = \{f\in \text{Homeo}(E):pf=p\}$. Let $\pi_1(...
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1answer
65 views

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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1answer
11 views

Equivalent Condition for Connectedness

I recently proved the following result after first proving it in $\mathbb{R}$ and generalizing: $\textit{Proposition:}$ Let $X$ be a $T_1$ space. Then $X$ is connected if and only if for every $a \in ...
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37 views

How do I prove that the compact open topology is metrizable?

Reference: Conway - Functions of one complex variable Let $G$ be open in $\mathbb{C}$ and $\{K_n\}$ be a sequence of compact subsets of $G$ such that $\bigcup_n K_n = G$ and $K_{n}\subset Int(K_{n+...
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1answer
32 views

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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1answer
55 views

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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37 views

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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11 views

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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1answer
13 views

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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1answer
16 views

Clouser of the set of a convergent sequence

Let $a_n$ be a sequence convergent to $q$ in $\bar U$ where $U$ is a bounded open set in $\mathbb R^m$. Let $B=\{a_n; n\in \mathbb N\}$. Why is $\bar B=B\cup\{q\}$ if $q$ in the boundary of $U$ and $\...
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1answer
20 views

A question about retraction mapping

Let $f:\overline{B}_r(0)\rightarrow \overline{B}_r(0)\setminus\{0\}$ be a continuous function. Suppose that $f(x)=x$, when $x\in \partial \overline{B}_r(0)$. Then we can consider a continuous ...
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1answer
25 views

Equivalent definitions for simple connectivity

For a path connected space $X$, it is simply connected iff any two paths sharing endpoints are homotopically equivalent. Here simple connectivity means it has trivial fundamental group. I'm doing ...
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2answers
34 views

Jordan curve in $C^2$

Can we find a Jordan curve $\gamma$ in $\mathbf{C}^2$ of class $C^1$ such that the projection to the first coordinate plane divides the plane into infinite components of connectivity.
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3answers
37 views

An uncountable subset of a second countable space has uncountably many of its limit points

Let $X$ have a countable basis , let $A$ be an uncountable subset of $X$. show that uncountably many points of $A$ are limit points of $A$. This is my attempt: By way of contradiction, assume that ...
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2answers
79 views

Alternative subbasis for the compact-open topology on $C(X,Y)$

Given spaces $X,Y$, where $X$ is Hausdorff, and the topology on $Y$ has basis $\mathcal{U}$, I would like to show that the set $\mathcal{S} := \{ S(K,U) \mid K \subset X, K \text{ compact, } U \in \...
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2answers
30 views

Determine if the following short exact sequence is split.

Do the following short exact sequences split? $0\longrightarrow A\longrightarrow B\longrightarrow \mathbb{Z}^2 \longrightarrow 0$ $0\longrightarrow\mathbb{Z}\longrightarrow A\longrightarrow B\...
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0answers
20 views

Compare the final (weak) topology and the box topology on $\mathbb{R}^\infty$

$\mathbb{R}^\infty$ is the space of sequences of real numbers $(r_1,r_2,\ldots)$ that are eventually $0$. That is, there's an $N$ so that $r_n=0$ for all $n>N$. We can consider this as a "limit" $\...
2
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0answers
22 views

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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2answers
33 views

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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1answer
19 views

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. ...
4
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1answer
31 views

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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6answers
641 views

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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2answers
22 views

How to finish? Connectedness [duplicate]

If $A$ is a connected set and $\{A_i : i \in I\}$, $I$ an arbitrary set (can be countable or not) of connected sets. How to show that if $A \cap A_i \neq \emptyset$ for all $i \in I$ then $A \cup (\...
2
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1answer
35 views

how to define a continuous function with special properties?

Let $X$ be a hausdorff topological space and $C$ be a closed proper subset of $X$ , how to define a continuous function $f:X\to\Bbb{R}$ such that : $f(x)=0$ for $x\in C$ $f(x)>0$ for $x\in X-C$ ...
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1answer
19 views

Explain a statement in a proof

At a proof at ProofWiki it is proved that there is a neighborhood $U$ of $x_0$ such as $fU\subseteq(c..d)$. In the proof it is used $(a): \qquad x \in {U_r}^- \implies f \left({x}\right) \le r$ I ...
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1answer
17 views

Proof Verification: C closed, convex, symmetric in Banach space X and $\cup_{n \in N \setminus 0} n.C= X$ then $B_\epsilon(0) \in C $.

I have an outline of the proof of this which I've expanded (correctly or otherwise) below, I'd appreciate feedback on it. (I think that C has to be closed in order to assert that $\cup_{n \in N \...
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1answer
30 views

$f: S^n \to S^n$ is the restriction of a continuous mapping $F: \overline{B^n_1} \to S^n$ iff $deg(f) = 0$.

this is a theorem from wikipedia (https://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping#Properties): A self-map $f: S^n \to S^n$ of the $n$-sphere is extendable to a (continuous) map $F: ...
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1answer
26 views

Closure of a set in the weak topology

Let $X$ be a Banach space, $S$ a subset of $X$. What is the closure of $S$ with respect to the weak topology?
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1answer
23 views

Applying topological definition of continuity to $f(x) = \frac{1}{x}$

I am trying to show that the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $$ f(x) = \left\{ \begin{array}{l} \frac{1}{x}, \, x > 0 \\ 0, \, x \leq 0 \end{array} \right. $$ is not ...
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4answers
74 views

Prove That $\mathbb{R}^n - \{0\} $ is connected for n > 1

I don't understand where to start proving this since $$\mathbb{R}^n - \{0\} = (-\infty,0)^n \cup (0, \infty)^n $$ Which is the union of two disjoint nonempty open sets, so it can't be connected. ...
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1answer
22 views

show set of matrices A such that I-A is invertible is open

If $U = \{A\in Mat(2,2) : I-A \text{ is invertible} \}$, how can I show that $U$ is open? I know that the set, say $V$, of $n\times n$ invertible matrices is open. Can I use this fact with the linear ...
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1answer
26 views

Topological space $X$ has no subbasis $S$ with property $Card(S)\leq Card(X)$

Let $X$ be a topological space and infinite which has no subbasis $S$ with property $Card(S)\leq Card(X)$. What special properties does it has? For example it's not metrizable. Because the ...
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1answer
46 views

Formula for number of faces in 4 dimensions

If a polytope has $m$ faces in 3 dimensions, how many faces does its analogous polytope have in four dimensions? Does a formula exist? For example, if $m=4$, you have a tetrahedron, and the 4-...
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0answers
29 views

$\mathbb{R}^J$ with box topology is completely regular

show that $\mathbb{R}^J$ in box topology is completely regular where $J$ is any set. I know it is completely regular in case uniform topology and box topology is finer than uniform topology. Therefore,...
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2answers
227 views

Boundary of $M_rp$ not equal to the sphere of radius $r$ at $p$?

My problem is to find a metric space in which the boundary of $M_rp$, where $M_rp = \{q \in M: d(p, q) < r \}$, is not equal to the sphere of radius $r$ at $p, \{x \in M: d(x, p) = r\}$. ...
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1answer
27 views

Why is the set $\{1/2\}\times(1/2, 1]$ not open in the ordered square?

Please refer to Example $3$ of §$16$ of Munkres for details. My attempt: $\{1/2\}\times(1/2, 1] = [1/2 \times 1/2, 1/2 \times 1]$, which is a closed interval, thus not open in $I \times I$. I don't ...
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2answers
83 views

Prob 6, Sec 7 in Munkres' TOPOLOGY, 2nd ed: The existence of an injection of a superset into the set means the sets have the same cardinality?

Let $A$ and $B$ be two sets such that $B \subset A$ and there is an injection $f \colon A \to B$. Then how to show that $A$ and $B$ have the same cardinality? Munkres' Hint: We define $A_1 \colon= ...
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1answer
101 views

Is there a general way to tell whether two topological spaces are homeomorphic?

We know that if two topological spaces $X$ and $Y$ are homeomorphic, then they have the same fundamental groups, and the same homology. In other words, we have functors $$\pi_1 : \mathsf{Top} \to \...
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1answer
42 views

Partial sum of bounded series is Cauchy in $C(X, \mathbb{R})$?

I am reading a proof of Tietze's Extension Theorem and there was a claim that, given a sequence of functions $h_n(x) : X \to \mathbb{R}$ If $$G = \sum\limits_{n = 1}^\infty h_n(x)$$ Is bounded, then ...
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1answer
38 views

Definition of second topological $K$-group of a Banach algebra

The question is a about the definition of the second topological $K$-group of a Banach algebra $A$. I was reading a text of Alain Valette (Prop. 3.3.7) where he proves that $$ K_1(SA) \cong \pi_1(\...
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0answers
38 views

Can we prove invariance of dimension directly from the Jordan-Brouwer separation theorem?

Is the following proof correct? Consider spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where $n<m$, and sphere $S^{n-1}\subset \mathbb{R}^n$. Suppose that we have a homeomorphism $f:\mathbb{R}^m \...
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0answers
22 views

Graph of a subset of $R^n$

I would like to know the definition of a graph of a subset of $\mathbb{R}^n$ which is a set. Each time, I found links about graph theory but here is my context:
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1answer
38 views

Embedding $T_{1}$-topological space in $\Bbb{R}^{J}$ [on hold]

Let $X$ be an infinite set and let $τ$ be a $T_{1}$-topology on $X$.Does there exist $J$ such that $(X,τ)$ can be embedded in $\Bbb{R}^{J}$ with product topology and cofinite topology on $\Bbb{R}$?...
2
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0answers
31 views

Necessary condition for the induced topology to be Hausdorff

Let $X$ and $Y$ be topological spaces and $f:X\to Y$ a continuous map. If $f$ is proper then $f(X)\subseteq Y$ is Hausdorff. However $f$ need not be proper for $f(X)$ to be Hausdorff. So what must $f$ ...
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1answer
26 views

Proper maps and their codomains

A continuous map $f:X\to Y$ is called proper map if for every compact $K\subset\subset Y$ the set $f^{-1}(K)$ is compact. Now, if $\mathbb D=\{z\in \mathbb C;|z|<1\}$. Why the map $f:\mathbb D\to ...
14
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1answer
886 views

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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0answers
49 views

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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1answer
21 views

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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0answers
23 views

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 ...