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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Question about bijective functions and homeomorphism

Is it true that "If two metric spaces each of which is the image of the other under a bijective continuous function, then the two metric spaces are homeomorphic."?? Thank you so much!!
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1answer
40 views

Is $C[0,1]$ locally Compact?

I'm asked to use the function $f_n(x)=nx$ for $0\le x\le \frac{1}{n}$ and $f_n(x)=1$ for $\frac{1}{n}\le x\le 1$. I'm not familiar with Functional Analysis.
2
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2answers
52 views

Strong version of Baire Category Theorem

We know that in a complete metric space or compact Hausdorff space the intersection of $\omega$-many open dense sets is dense. In such spaces is the intersection of fewer than $2^\omega$-many open ...
5
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1answer
83 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
5
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2answers
196 views

Is second-countability invariant under homotopy equivalence?

I am wondering if second-countability is invariant under homotopy equivalence. If I had to guess I would say so. Intuitively, if we have a countable basis of a space, and then stretch, contract, bend ...
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1answer
16 views

On the wording of a question related to open cover of sets

I am working on the exercise where the hypothesis is : Let $X$ be a scheme such that there exists affine open subsets $U_i \ (1 \leq i \leq n)$ such that $X = \cup U_i$. Further any two of the $U_i$'s ...
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0answers
40 views

Retracts of $\mathbb{Q}$

Let $A \subseteq \mathbb{Q}$ be a non-empty set. Is $A$ a retract of $\mathbb{Q}$? In other words, is there a continuous map $r: \mathbb{Q}\to A$ such that $r|_A = \mathsf{id}_A$?
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1answer
30 views

Show that for a degree 1 map $f: M \rightarrow N$ the induced map $f_*: H_1(M) \rightarrow H_1(N)$ is a surjection

I'm trying to solve the following problem: Show that for a degree 1 map $f: M \rightarrow N$ of connected, closed, orientable manifolds, the induced map $f_*: \pi_1(M) \rightarrow \pi_1(N)$ is ...
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1answer
30 views

$\lim_{x \to +\infty} f(x), \lim_{x \to -\infty} f(x)$ both exist and are finite, $f$ uniformly continuous

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous and suppose that $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ both exist and are finite. How do I show that $f$ is uniformly continuous? ...
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0answers
30 views

Homeomorphism between $S^1\times \Bbb{R}$ and $\Bbb{R}^2\setminus\{(0,0)\}$

I need to show that there exists a homeomorphism between $S^1\times \Bbb{R}$ and $\Bbb{R}^2\setminus\{(0,0)\}$ If we map $(\cos x, \sin x,y) \to (y\cos x,y\sin x)$ then only problem is that it is ...
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1answer
1k views

Neighborhood Base (the definition)

In Steven G. Krantz' A Guide To Topology, a countable neighborhood base is defined: Let $(X,U)$ be a topological space. We say that a point $x\in X$ has a countable neighborhood base at $x$ if there ...
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0answers
18 views

Subspace which is an image, but not a retract [closed]

Is there a topological space $(X,\tau)$ and a non-empty subset $S\subseteq X$ such that there is a surjective continuous map $f:X\to S$, but $S$ is not a retract of $X$ (there is no surjective map ...
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0answers
27 views

Importance of Urysohn Metrization Theorem

I was just wondering if there are good examples that stem from the Urysohn Metrization Theorem. Are there any concrete examples that we use in Calculus or Analysis that stem from the result? I just ...
1
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0answers
27 views

No norm consistent with given topology

Given the (Frechet) topology on the Schwartz class $S(\mathbb{R}^d)$ induced by the seminorms $\rho_{\alpha \beta}f = \operatorname{sup}_{x \in \mathbb{R}^d}|x^{\alpha}\partial^{\beta}f|$, how can I ...
1
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1answer
38 views

Unit sphere weakly dense in unit ball

I'm studying for an exam and came across a problem: I want to prove that the unit sphere in a Hilbert space $\mathcal{H}$ is weakly dense in the unit ball. I already had to prove that the unit ball ...
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0answers
19 views

Hilbert cube in metric Spaces

Is there proof that does not use the concept of " product space " to prove that the Hilbert cube is closed and bounded ?
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2answers
41 views

A problem in topology relating to the finite intersection property

This is a problem from Munkres' Topology Exercise 37.1 (c) Let $X$ be a space. Let $\mathscr{D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. ...
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0answers
18 views

A problem of a collection of subsets maximal with respect to the finite intersection property [duplicate]

Let $X$ be a space. Let $\mathscr {D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. Show that if $X$ is Hausdorff, there is at most one point ...
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0answers
37 views

If $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$

I want to show that if $X$ and $Y$ are homotopic and $X$ is contractible, so is $Y$. It feels like I'm missing something really obvious. $X$ is homotopic to $Y$, so there exists $f: X \to Y$ and $g: Y ...
4
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1answer
31 views

connected space of matrices

Let $M(n, \mathbb{R})$ be a set of square matrices. Consider the subset $S$ of $M(n, \mathbb{R})$ where the absolute value of the eigenvalues of the matrices in the subset are $\le 2$. Is this subset ...
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3answers
59 views

Complete metric space, not simply-connected

I've been going over the algebraic topology part of Munkres and this question has stumped me. If we have a complete metric space that is not compact, must it be simply-connected (path-connected plus ...
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1answer
40 views

Nature of Equilibrium Points

I would like to prove the following: "The nature of the equilibrium points (i.e. stability/instability) of a one-dimensional differential equation remains invariant under the effect of the ...
2
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0answers
30 views

A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
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1answer
43 views

Counterexamples to Brouwer’s fixed point theorem

Brouwer’s fixed point theorem states that for any compact convex set $X$, a continuous mapping from $X$ to $X$ has at least ones fixed point. If we replace the convex condition with let's say ...
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1answer
37 views

general topology- Ordinal spaces [closed]

I need some help with my set theory homework if anyone is willing to offer some solution I'll be more than gratefull. 1) If $A$ is a subset of the ordinal space $[0,\alpha[$ then the cardinality of ...
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1answer
26 views

Kuratowski's definition of a topological space, and immediate consequences

According to Kuratowski, a topological space is a set $X$ together with an operation ${\bf C}$ (called closure) which associates to every element $A\subset X$ an element ${\bf C}A\subset X$ such that: ...
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4answers
388 views

Can a 2D person walking on a Möbius strip prove that it's on a Möbius strip?

Or other non-orientable surface, can a 2D walker on a non-orientable surface prove that the surface is non-orientable or does it always take an observer from a next dimension to prove that an entity ...
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1answer
23 views

A space is completely regular if and only if it is homeomorphic to a subspace of $[0,1]^ J$ for some $J$

This is a theorem from Topology by James Munkres. Theorem 34.3 A space is completely regular if and only if it is homeomorphic to a subspace of $[0,1]^ J$ for some $J$ The book merely states that ...
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1answer
35 views

A metric and discrete topology

Let $\Sigma=\{1,2,...,n\}$ and $\Omega=\Sigma^\mathbb{N}$ be the set of infinite sequence of n digits. Define a metric $d$ on $\Omega$ by $d(\omega,\tau)=2^{-|\omega\wedge\tau|}$ where ...
2
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1answer
22 views

Is a closed map from X to T1 proper?

$f:X→Y$ is a closed map. Suppose that the inverse image of each point in $Y$ is a compact subset of $X.$ Show that $f^{-1}(K)$ is compact in $X$ whenever $K$ is compact in $Y.$ This is my homework ...
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0answers
85 views

A question about compact sets: how to prove $g$ must be an isometry

Let $(X,p)$ be a compact metric space. Suppose that $g:X\rightarrow X$ is a function such that for all $x_1,x_2\in X$ we have $p(g(x_1),g(x_2))\geq p(x_1,x_2)$. Prove that, in fact, $g$ must be an ...
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2answers
27 views

Is $(-3n,3n)$ a subcover of $(-n,n)$?

I read this: Let $C=\{(-n,n):n \in \mathbb N\}$ and $C'=\{(-3n,3n):n \in \mathbb N\}$. Then $C$ and $C'$ are open covers of $\mathbb R$ (understood) and $C'$ is subcover of $C$ (not understood). A ...
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1answer
12 views

Does the Function Between Some Set and N Has Any Properties? (Question about Countable Set)

(Note that when I say countable, it means countable infinite) My question is that, the definition of a countable set is $S$ is said to be a countable set if there is a bijective function ...
2
votes
1answer
161 views

question related to perfect maps preserving compactness

A perfect map $f$ is a closed continuous surjective function such that the preimage of every point is compact. One property of perfect maps is that if $f \, \colon \, X \to Y$ is perfect, and $Y$ is ...
3
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0answers
44 views

Is the distance induced by the topology of the set a metric

First of all excuse me if something with the question is wrong. I am not very knowladbable in this area, though I know what I am asking and my question should be infact a reasonable one. The topology ...
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1answer
38 views

How to build a covering space ?? [closed]

I have to build a covering space $ p:\mathbb{R} ^{2} \rightarrow S^{1} \times S^{1}$ How to do it?
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1answer
85 views

Are the following topological spaces locally compact?

I am trying to determine whether the following spaces are locally compact: a) the slotted plane b) the radial plane For part a) I am almost certain that it is not compact, but not sure how to go ...
0
votes
1answer
31 views

limit points of $[0,1]$

$[0,1]$ defined in $\mathbb{R}$ is a closed set. A closed set contains all its limit points. The points $0$ and $1$ do not have all their neighbourhoods within $[0,1]$. So they are not limit points ...
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0answers
26 views

Properties of Hilbert Spaces- Contrasting Two Different Topological Spaces

Let H be the space of real sequences x = $(x_1 , x_2, ... )$ with $\sum(x_n^2)$ finite. (This is $l_2$ in fact.) I wish to show the following: The topology on H is ...
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1answer
15 views

0 limit point of spectrum of completely continuous operator $H\to H$

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 475 here) that 0 is an accumulation point for the spectrum of a completely continuous operator $A:H\to H$ where $A$ ...
0
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1answer
20 views

Prove a given set on the euclidean plane is connected

Let $f:\mathbb{R}\longrightarrow\mathbb{R},\ g:\mathbb{R}\longrightarrow\mathbb{R}$ be functions, that satisfy $f(x)\leq g(x)\ \forall x\in\mathbb{R}$ and let $S=\{(x,y)\in\mathbb{R}^2:f(x)\leq y\leq ...
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2answers
404 views

Proof of Pasting Lemma

I am trying to understand the proof of the Pasting Lemma. I have found several proofs but I am missing something from all of them. Wikipedia has: Statement: Let $X,Y$ be both closed (or both open) ...
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1answer
45 views

Paracompact and countably compactly generated

A space X is countably compactly generated if it can be written as countable direct limit of compact Hausdorff spaces. Are countably compactly generated spaces paracompact spaces? Do we have ...
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2answers
48 views

How to show the covering space of an orientable manifold is orientable

I'm trying to prove this using purely topological arguments, no differential geometry as I haven't been exposed to it. I've been playing around with definitions a bit and here's what I have so far. ...
2
votes
1answer
70 views

Is $\mathbb{A}^1$ isomorphic to one of its quasi-affine subsets?

It's well known that $\mathbb{A}^1$ is not isomorphic to any proper open subset of itself. Just out of curiosity, is $\mathbb{A}^1$ isomorphic to any proper quasi-affine subset of itself, or is this ...
2
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0answers
48 views

An example of finite, connected topological group

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think ...
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0answers
19 views

Question related to Uniform Space

I have questions related to Uniform Space; If $X$ is a countable discrete space, then how to show that finest pre compact uniformity on $X$ admits a countable base of entourages. If $\mho$ is a ...
2
votes
1answer
49 views

A question (more like three) about a topological space of ordinals.

I've been struggling with these for a while now, if anyone is willing to offer a hint I'll be more than grateful. Given an ordinal $\varepsilon$, consider the topological space $L_{\varepsilon}$ ...
2
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3answers
100 views

$M \times N$ orientable if and only if $M, N$ orientable

For two manifolds $M$ and $N$ I'm trying to prove that $M \times N$ is orientable if and only if $M$ and $N$ are orientable. My attempt so far: $\impliedby)$ Assume $M, N$ are orientable. Then ...
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1answer
41 views

How to draw a quotient space

I'm trying to understand this exercise: What does the author mean? I could find proprieties of this quotient space and everything but I don't know how to draw it, besides that it seems there are a ...