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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39 views

Closure of a bounded set

I have troubles with proving that the closure of a bounded set is bounded. Can somebody help me out with this problem? Thanks!
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97 views

Topology of $GL_n(K)$

I need to show any of the following results: Consider $K=\mathbb{R}$ or $\mathbb{C}$, then, 1) The compact-open topology and the usual topology of $GL_n(K)$ are the same. 2) Taking inverses and ...
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72 views

Not equicontinuous

I am trying to show that if a family of continuous functions $\mathcal{F}$ in $C(X)$ is uniformly bounded and is not equicontinuous, then there exist a sequence $(x_n) \in X$ convergent to $x_0$ and a ...
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42 views

A sequence of neighbourhoods with decreasing Haar measure in a non-discrete group

Let $G$ be a non-discrete (LCH) group. How can one find a sequence $(V_n)$ of compact, symmetric neighbourhoods of the identity element $1$ such that $\mu(V_n)\to 0$, where $\mu$ denotes the Haar ...
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71 views

Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function

In Walter Rudin's Principles of mathematical analysis Exercise 5.21, it is proved that for any closed subset $E\subseteq \mathbb{R}$, there exists a smooth function $f$ on $\mathbb{R}$ such that ...
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36 views

discrete metrizable space

Im studing topological spaces $X$ with the following property If $Y\subseteq X$ is such that $|Y| =|X|$ then $X\cong Y$. Additionaly, suppose that such $X$ is a metric space, I want to prove that ...
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54 views

Basis of an infinite dimensional Banach space

Can somebody please check the correctness of this proof, since I am new to this? Thank you in advance. Given $X$ a normed space and $Y$ a proper subset of $X$ that is a linear subspace, prove that ...
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26 views

How to check if this function is semicontinuous

Could you tell me how to check that this functions are semicontinuous? $(X, \tau)$ - topological space, $ \ X \neq \emptyset$, $ \ f: X \rightarrow \bar{\mathbb{R}}$, $ \ \bar{\mathbb{R}} = [- ...
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48 views

If J is uncountable, then $R^J$ is not normal.

If J is uncountable, then $R^J$ is not normal. Let X = $(Z+)^J$; it will suffice to show that X is not normal, since X is a closed subspace of $R^J$ . We use functional notation for the elements ...
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29 views

Find a subspace homeomorphic to the quotient.

I'm giving the equivalence relation $V\sim W$ when $\exists\lambda>0:\lambda V=W$. I have to prove that $\sim$ is an equivalence relation and find a subspace of $\mathbb{R}^2$ homeomorphic to the ...
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100 views

Are these quotient spaces homeomorphic to a cylinder and to the Möbius Strip?

Consider for $[0,1]\times [0,2]$ the function $f:\{0\}\times [0,1]\to \{1\}\times [0,1]$ given by $f(0,x)=(1,x)$. Prove that the quotient space given by this $f$ is homeomorphic to the cylinder and ...
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97 views

Topology: Open, Closed Set and infinity norm

Since last week I've been learning a bit about Topology in Calculus and know the basic definitions of open, closed, norm, etc. Now I try to solve this question but I don't know how to. Its really ...
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77 views

Topology. Why is $T^{-1}$ continuous?

Today we did this proof, but we could not finish it and our prof said that the end would be easy, but I could not finish this proof. Let $X$ be a $T_3$ space with a countable basis $B$. Then we ...
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43 views

Does there exist a subset $S\subset\mathbb R$ such that inf $\{a>0:S+a=\mathbb R-S\}=0$?

I founded the following question a good challenge in real analysis and topological properties of real line... Does there exist a subset $S\subset\mathbb R$ such that inf $\{a>0:S+a=\mathbb ...
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38 views

Union of a sequence of path connected sets being path connected

Theorem: Let ${A_n}$ be a sequence of path connected subsets of a space $X$ such that for each integer $n\ge 1$, $A_n$ has at least one point in common with one the preceding sets $A_1, ..., A_{n-1}$. ...
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3answers
243 views

Showing that the function $C(b)$ is a compact set for $|b| > 1$

I am reading "An Invitation to Dynamical Systems", and one of the challenge problems is to prove that $C(b)$ is a compact set where $C(b)$ is defined as the set of all numbers that can be expressed in ...
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40 views

Assigning manifolds to graphs in a functorial way

I am looking for ways to functorially assign manifolds (or more general topological spaces) to families of graphs. To be more precise, I am interested in functors from specific subcategories of the ...
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46 views

$\{{(x,K) : x \in K}\}$ is closed in $X \times \{\text{closed subsets of }X\}$

Let $X$ be a locally compact (and Hausdorff, if needed) space. Let's denote the closed subsets of $X$ by $C(X)$, equipped with the Hausdorff topology. I want to prove that $F=\{{(x,K) : x \in K , K ...
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24 views

Openess of sets given by equivalence relations in the quotient topology.

I'm trying to prove this: Let $R$ be an equivalence relation in $X$. Show that $A$ is open in $X/R .\iff \bigcup_{[x]\in A}[x]$ is open in $X$ One of the first things that come to my mind is ...
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3answers
58 views

What is a “control point”?

I'm trying to figure out a good definition of control point for use in wikipedia (see https://en.wikipedia.org/wiki/Control_point_(mathematics) ) There seems to be a bias towards ascribing a ...
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90 views

Completeness of the metric r(x,y)=min{d(x,y},1}

Can somebody help me out with the following: If (X,d) is a metric space and r(x,y)=min{d(x,y},1} for all x,y in X. I proved that r is a metric and that r and d are equivalent. Now I want to prove ...
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95 views

prove that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$ w.r.t these metrics

Prove that $\mathbb{Q}^n$ is dense in $\mathbb{R}^n$ w.r.t these metrics: $d_{2}(x,y)=(\sum_{k=1}^{n}(x_{k}-y_{k})^2)^\frac{1}{2}$ $d_{0}(x,y))=\max_{1\leq k\leq n}|x_{k}-y_{k}|$ my attempt of ...
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49 views

B meager, not empty, not open implies complement is dense

Let X be topological space. The subset B is meager, not open and non-empty. How can you prove that the complement is dense?
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98 views

homeomorphism between a compact and a Hausdorff space [duplicate]

As I was going over my lecture notes I found the following problem which we haven't proof in class In the example we know $H$ is a Hausdorff topological space and $C$ is a compact topological space ...
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57 views

Are these two definitions equivalent?

If $\langle A_n : n \in \omega \rangle$ is a sequence of subsets of a set $X$, $$ \underline{Lim} A_n = \{ x \in X : \exists n_0 \in \omega \forall n \geq n_0, x \in A_n \} $$ If $\mathcal A$ ...
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21 views

Help undestanding compactness with convergent subsequences

One way to define compactness in metric spaces is to note that in compact metric space each sequence has a convergent subsequence. Understanding compactness is difficult for me from this ...
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43 views

Proof: In a topological vector space, every neighborhood of $0$ contains a balanced neighborhood of $0$

I was reading this proof in Rudin 2/e (Th 1.14), but couldn't work it out. Suppose $U$ is a neighbhorhood of $0$ in the topological vector space $X$, then Since scalar multiplication is ...
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123 views

Is $A = f\mathbb{R^2}$ complete? $f:\mathbb{R^2}\to\mathbb{R^3}$, $f(x,y) = (x,y,x^2-y)$

My question is pretty much what it says in the headline. Is $A = f\mathbb{R^2}$ complete, where $f:\mathbb{R^2}\to\mathbb{R^3}$, $f(x,y) = (x,y,x^2-y)$, $(x,y)\in\mathbb{R^2}$. $f\mathbb{R^2}=$$\{ ...
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37 views

Dugundji problem about quasicomponents.

Problem 5 part d), of chapter V, section 3 (p. 118): d. In $E^2$ let $L_1$ be the line $x=1$ and $L_2$ the line $x=-1$. For each $n\in\Bbb Z^+$ let $R_n$ be the rectangle $\{(x,y):|x|\le ...
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88 views

What path to get to topology?

I am in calc 1 right now and was wondering what kind of journey is ahead of me before topology. I really want to study high level math like this but am not sure if I want to major in math. I am a pre ...
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127 views

Example of closed and bounded set that is not compact

Consider the metric space $Q$ of rational numbers with the Euclidean metric of $R$. Let $S$ consist of all rational numbers in the open interval ($a, b$), where $a$ and $b$ are irrational. Then $S$ is ...
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43 views

Confusion about definition of quotient topology

I have some confusion when I learn about the definition of quotient topology. Define $\sim$ on $[0, 1]$ by $s ∼ t$ if and only if $|s − t| = 0 $ or $ 1 $ Let $q$ be the quotient map and $$T_q =\{V ...
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45 views

What does it mean that the exponential topology of a space is T$_1$?

If the exponential topology $\exp(X)$ is $T_1$, does that mean that for every $x \in X$ we have that $\{x\}$ is closed in $\exp(X)$ or does that mean that $\{\{x\}\}$ is closed or does it mean ...
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49 views

Hausdorffness of $\exp(X)$ implies regularity of $X$

I have trouble proving this fact about the exponential topology If $\exp(X)$ is Hausdorff and $X$ is T1, then $X$ is T3. What I need to show is that if I have a (i.e. closed) $\{x\} \subset X$ ...
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49 views

Help understanding a proof

So the textbook I'm reading just stated three definitions of continuity: a) f is continuous at a (using neighborhoods) b) the epsilon-delta definition c) If $x_n$ is a any sequence of elements of ...
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29 views

I need help with understanding this proof that the product topology on $\mathbb{R}^n$ is the same as that induced by the square metric

Here's the proof: I'm basically struggling with following the logic used in the proof. How is for each $i$ there an $\epsilon_i$ such that $(x_i - \epsilon_i, x_i + \epsilon_i)$ is contained in ...
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26 views

Proof regarding compact $T_2$-spaces and closed continuity.

Question: Prove that any continuous function from a compact $T_2$-space onto a $T_2$-space is closed, that is, $f(F)$ is closed if $F$ is closed. Is my general reasoning correct? Any compact subset ...
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45 views

Compactness in $R^n$

I am looking at the proof that the following statements are equivalent from Apostol's Mathematical Analysis. Let $S$ be a subset of $R^n$. b) $S$ is closed and bounded. c) Every infinite subset of ...
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66 views

Subspaces and convergence in weak* topology

I would like to ask some questions regarding convergence in the weak* topology and subspaces. Let $X$ be a normed space with subspace $A \subset X$. Assume $X$ is endowed with the weak* topology. ...
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60 views

The interval $(0,\infty)$ is an open set.

I want to prove this using interior points, $\epsilon$-neighborhoods and interior sets. The interior of a set A is denoted $A^o$. To show that $(0,\infty)$ is an open set, we must show that ...
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36 views

Is a mapping f bijection ?

If Y is an one point compactification of X,Y=X union {p}, p not belong to X. Is a mapping f from X into Y bijection? If it is not, what are the assumptions I add to be f bijection ? Thanks for any ...
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77 views

weak-* topologies

Say $S = \{z \in \ell_\infty : z_n \in \{0,1\}\}$. Suppose I am asked a question about the weak-* topology on $S$. How am I supposed to make sense of this? The weak-* topology is a topology on a dual ...
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154 views

Dense sequence in $[0,1]$

There is the theorem proved by Liouville which states that if $x$ is irrational then there are infinitely many fractions $\frac{p}{q}$ such that $|x-\frac{p}{q}|<\frac{1}{q^2}$, i.e. ...
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48 views

Sequence of increasing compact sets

Suppose $X$ is a locally compact metric space which is $\sigma$-compact. Let $K$ be a compact subset of $X$. We can find a sequence of compact sets $K_{n}$ such that $K_{n} \subset \textrm{int}(K_{n + ...
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47 views

Existence of a neighbourhood of a compact set ( from james fibrewise topology)

I'm reading James' Fibrewise topology book and I'm trying to understand the proof of proposition 7.4 , it says: Let X be a proper G-space . Then X is fibrewise regular over X/G. Proof For any $x \in ...
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31 views

Show that any infinite set $X$ may be endowed by a metric d such that $X$ has a limit point in $(X,d)$

This is an exercise I've been dealing with for a few days; I was wondering if anyone could help me with a hint or just telling me the answer. Regards
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92 views

Fundamental Group of Klein Bottle?

Let $C^{*}=C \setminus \{ 0 \}$. What is the fundamental group of $C^{*}/H$, here $H=\{\psi^n;n \in \mathbb{Z}\}$ with $\psi(z) = 2 \bar{z}$?
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60 views

Generalisation of Tietze Extension Theorem for Compact Hausdorff Spaces

Let $X$ be a normal space and $A$ a closed subspace of $X$. Let $Y$ be a compact Hausdorff space. Is there a theorem that allows any continuous $f : A \rightarrow Y$ to be extended to a continuous $F ...
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94 views

Show that if $h,k: S^1\rightarrow S^1$ are homotopic, they have the same degree.

We define the degree of a continuous map $h: S^1 \rightarrow S^1$ as follows: Let $b_0$ be the point $(1,0)$ of $S^1$; choose a generator $\gamma$ for the infinite cyclic group $\pi_1(S^1,b_0)$. If ...
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29 views

Is this an example of a sequential non-Fréchet–Urysohn space?

Let $X$ be the set $X = \{ (0,0) \} \cup \{ (\frac{1}{n},0) : n \in \mathbb N \} \cup \{ (\frac{1}{i},\frac{1}{k}) : i,k \in \mathbb N \}$. Points of the form $(\frac{1}{i},\frac{1}{k})$ are ...