# 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; ...

303 views

### The continuity of a distance function

Let $(X,d)$ be a metric space $A\subset X$ be a nonempty subset. The distance function $f : X \to\mathbb R$ by $f(x)=d(x,A)$ where $d(x,A) = \inf_{a\in A} d(x,a)$ and $\mathbb{R}$ denotes the set of ...
136 views

### Is my general approach to proofs acceptable? A general topology example.

Proving: $A$ is closed iff $A = \bar{A}$. "To the right": If $A$ is closed, $A = \bar A$ If $A$ is closed this means that it contains all of its own accumulation points. And we would find that its ...
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### Uniformity is generated by pseudometrics

How to prove that every uniform space is generated by a family of pseudometric spaces? You may offer me a book. In Engelking this theorem is presented without a proof. In Willard it is a exercise. ...
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I am trying to find a counter-example related to the definition of paracompactness, but it seems that it is not very easy. Here is the problem. Give an example to show that if $X$ is paracompact, ...
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### Intuition behind continuity in topological spaces

I was approaching the following problem: "Let $f \colon X \to Y$ be continuous. Is it true that if $x$ is a limit point of $A \subset X$ then $f(x)$ is a limit point of $f(A)$?" The answer is that ...
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### Hartshorne II Prop 2.6

Prop 2.6 constructed a continuous map $X$ to $t(X)$, I cannot verify that it is a homeomorphism. I try to show any open set $U$ is mapped to $t(X)\setminus t(X\setminus U)$. To show it is surjective, ...
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### Looking for articles on postcritically finite rational maps in Russian or French

I'm looking for articles on postcritically finite rational maps. I found a few articles in English, but I can't find any in Russian or French.
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### Is this case possible (hedgehog metric, colinearity)

My topology class was asked to prove that the hedgehog metric was indeed a metric (the details are irrelevant for my question). This does not concern the proof itself, but rather the structure of the ...
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### Not 1-dimensional homological equivalent of the circle

The questions origins from this problem and my incorrect answer to it. I'm trying to correct it, but it turned out that the topological space - that I need to do it straightforward - has very specific ...
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### non-trivial convergent sequence

In the Stone-Čech compactification $\beta X$, there is no non-trivial convergent sequence. Assume $R=(\mathcal{U}_{n})_{n}$ is a sequence of distinct ultrafilters on some set $X$. Since every ...
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### (Regular (but not Lindelöf)) + (??some property??)=> Normal

Does anyone know any sufficient condition to a regular topology (but not Lindelöf) to be normal?
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### What is the relation between singular point for a function and the one in a vector field?

What is the difference between sigular point for a function and the one in a vector field? Is the derivative or divergence at the singular point must be infinity? By the way, what is the relation ...
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### Is there a $P$-space linearly Lindelöf and non-Lindelöf?

A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersection). A topological space $X$ is linearly ...
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### Is this proof correct: domain of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected.

The domain $X$ of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected. Proof: If $X = F \uplus G$ for two nonempty closed sets $F,G$ ...
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### Cantor set homeomorphism [duplicate]

I am trying to prove that $\{0,1\}^\mathbb{N}$ is homeomorphic to the cantor set. Consider the mapping $f:\{0,1\}^\mathbb{N}\to[0,1]$ defined as$$f(x)=2\sum_{n=0}^\infty 3^{-n-1}x(n)$$ I think that ...
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### Union of Sets in Locally Compact Hausdorff Space

Is it possible for an open set in a locally compact Hausdorff space to not be the union of an increasing sequence of compact sets? If so, given a regular Borel measure on such a space, how is it that ...
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### example path but not connected [duplicate]

In my course notes of topology, we have seen the following example of a space that is connected but not path-connected. Define $X := \{(0,0)\} \cup \{(x,\sin(\frac{1}{x}))\mid x > 0\}$ I ...
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### Piecewise continuous selection from a correspondence

Take a correspondence $C:[0,1] \rightarrow [0,1]$ which is non-empty, convex valued and has closed graph. For each $x \in [0,1]$ let $a(x) = \{ \max \ y\in C(x), \min \ y \in C(x) \}$ (i.e., the set ...
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### How does the base in the complete lattice from a given topology look like?

I been reading back and forth about lattices and topologies all day. And one thing I can't seem to get a good idea about is how a base would look in such a complete lattice. And what would be the ...
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### intro. to topology mendelson - closure in a subspace

I'm self studying intro to topology by Mendelson and I just completed a book problem and wanted to get input on whether it's okay. The problem statement is, Let $Y$ be a subspace of $X$ and ...
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### How to show that the distance between these sets is positive?

Let $T_i=\{(1-t)x_i+ty_i;\;0\leq t\leq1\}$, where $x_i,y_i\in\mathbb{R}^n$; $i=1,2$. Could someone help me to prove that if $x_2= (1+\varepsilon)x_1$ and $y_2= (1+\varepsilon)y_1$ for some ...
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### upper semi-continuity of a multi-valued function $T$ and lower semi-continuity of $d(x,T(x))$

Let $(X,d)$ be a complete metric space, $CB(X)$ the set of closed and bounded subsets of $X$, and $T:X\rightarrow C(X)$ be a multi-valued function. How can you prove this: If $T$ is upper ...
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### Covering argument

In proving Harnak's inequality (I am referring to this article: "On Harnack’s Theorem for Elliptic Differential Equations"Communications on Pure and Applied Mathematics Volume 14, Issue 3 ), Moser ...
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### Using Minimax Theorem of prove the determinacy of closed games?

In this paper, page 6, Itai Arieli and Yehuda Levy mention briefly using using Minimax Theorem of prove the determinacy of closed game in a more general setting. The minimax theorem they mentioned ...
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### Check my answer: H is compact $\iff$ every cover {${E_\beta}$}$_{\beta \in A}$ of $H$ has a finite subcovering.

Question: Let $H ⊆ \Bbb R^n$ Prove that H is compact $\iff$ every cover $\{{E_\beta}\}_{\beta \in A}$ of $H$, where $E_\beta$ 's are relatively open in $H$, has a finite subcovering. Solution: ...
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### Dimension theory “based on $\mathbb R^n$”

This question is somewhat vague, so please be gentle with me. I want to know if there is some definition of topological dimension that has $\mathbb R^n$ as a "paradigm", something like 'A nice ...
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### Infinite collection of homeophisms from $S^1\times S^1$ to $S^1\times S^1$ that are not homotopic.

While studying for my topology exam I tried to come up with a solution of the following problem. Let $X=S^1\times S^1$. Give (with proof) an infinite collection of homeomorphisms $f_i:X\to X$ such ...
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### If $E_1, E_2$ are connected and $A_1\subseteq E_1$ and $A_2\subseteq E_2$ then $(E_1\times E_2)-(A_1\times A_2)$ is connected [duplicate]

Let $E_1$, $E_2$ connected metric spaces. Let $A_1\subseteq E_1$ and $A_2\subseteq E_2$ proper subsets. Show that the complement of $A_1\times A_2$ $$(E_1\times E_2)-(A_1\times A_2)$$ is connected. I ...
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### Intersection form on manifolds with boundary

It is a "basic fact" that the intersection form of a closed oriented 4k-dimensional manifold is unimodular. (Could anyone point me to a reference to a proof of this fact?) What can be said about the ...
1k views

### The complement of every countable set in the plane is path connected [duplicate]

I'm trying to show that if $A$ is a countable subset of $\mathbb{R}^{2}$ then $\mathbb{R}^{2}\backslash A$ is path connected. I already have the idea of the proof. It is clear to me that for every two ...
363 views

### If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact

Let $X$ be a Banach space and $C\subset X$. $\fbox{1}$ If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact ? $\fbox{2}$ If $C$ is convex , weakly-closed ...
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### Fibers and they being discrete space

When talking about fibers in topology and geometry, especially in fiber bundle, vector bundle and fibration, it is said that they are discrete spaces. But in some diagrams I see, they are often ...
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### How to prove this manifold is a sphere?

Suppose $M$ is a compact topological manifold, and it is covered by two open set $A$ and $B$, both homeomorphic to $n$-dimensional discs. How can I show that $M$ is homeomorphic to a $n$-dimendional ...
Let $X$ be a topological space that satisfies the following condition at each point $x$: For every open set $U$ containing $x$, there exists an open set $V$ with compact boundary such that $x\in ... 0answers 53 views ### Topological graphs Given the universel covering space$\hat{X}$of$X$by$p:\hat{X}\rightarrow X$, there exists a bijection between subgroups$H<G=\pi_1(X,x_0)$and covering spaces$\tilde{X}\rightarrow X$with ... 0answers 53 views ### field lines terminating at infinity A dipole consists of two equal and opposite point charges separated by a fixed distance. With two exceptions, all the electric field lines begin on one charge and end on the other. In the two ... 0answers 60 views ### A question on semi-stratifiable spaces A space$(X,\tau)$is called semi-stratifiable if there exists a function$g:\omega\times X\to\tau$such that: (i) for any point$x$of$X$holds$\{x\}=\bigcap_{n\in\omega} g(n,x)$; (ii) for any ... 0answers 51 views ### Jordan curves, its interiors and the existence of a continuous function. Let$L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$is a Jordan curve,$O(t)$is its interior and$H(t,s)=L_t(s)$. If$H$is a homotopy from$L_0(s)$to$L_1(s)$, is there exists a continuous ... 0answers 152 views ### a problem on metric spaces I am reading the book by Burago and Ivanov "A course in metric geometry". I tried to do some problems but have some difficulties. For example, page 66 exercise 3.1.26: Let$(X, d)$be a metric space ... 0answers 101 views ### Compute$df_1: ST_1^3 \rightarrow TSO(3)_I$In short, the problem is to compute$df_1: T_1S^3 \rightarrow T_{I}SO(3)$, given$f: S^3 \rightarrow SO(3), r \in S^3, f(r) \in SO(3): f(r)(q) = rqr^{-1}, q\in R^3$. I just get to study differential ... 0answers 126 views ### Problems about continuity of$|f|$and$f\vee g$; confusion about definitions I can't seem to wrap my head around this notation of my textbook can some please explain to me what this says? What I am trying to show? (a) Given$f: D \to \mathbb {R}$, let$|f|$be the ... 0answers 86 views ### Following a polyline along the surface of a polygon that is twisted I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ... 0answers 107 views ### Cantor set as a set of continued fractions? Does the classical cantor set have a nice description as a set of continued fractions? I made a (superficial) search and didn’t find anything, but I’m very tired right now, so please forgive me that ... 0answers 196 views ### Show that the projection map is Orientation preserving iff n is even My question is that Orient the unit sphere$S^n$in$\Bbb R^{n+1}$as the boundary of the closed unit ball. Let$U$be the upper hemisphere$U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ... 0answers 49 views ### Orientation-preserving diffeomorphism [duplicate] Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question. 1answer 78 views ### Is$Y$open in$X\cup_f Y$? Let$X,Y$be topological spaces,$A\subset X$- a subspace and$f:A\rightarrow Y$- a continuous map. Then we can define$X\cup_f Y = X\sqcup Y/\{a\sim f(a)\quad a\in A\}$Then the composition ... 1answer 151 views ### Why are the real numbers with the K-Topology not metrizable? I will denote the real numbers with the$K$-Topology as$R_{K}$(If someone doesn't know or remember this topology, read here). I understand that$R_{K}$is not regular, since the set$K$cannot be ... 1answer 123 views ### I'm mixed up in my Definitions for a closed and open Set in Topology I tried to understand topology recently and I'm confused on what is considered open and closed in Topological sets/spaces. Say you have a function where there exists some ... 0answers 79 views ### A question on star$\sigma$-compact spaces A topological space$X$is said to be star$\sigma$-compact if whenever$\mathscr{U}$is an open cover of$X$, there is a$\sigma$-compact subspace$K$of$X$such that$X = ...
It is well known that if $X$ is a metric space then sequentially compactness and compactness are equivalent.\ Now we consider a normed vector space $E$ and its dual $E^\ast$. From Banach ...