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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2
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3answers
34 views

A Compact Hausdorff Space with no Manifold Structure?

What is an example of a compact Hausdorff space that cannot be given the structure of a (i) differential manifold (ii) topological manifold?
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0answers
12 views

When do two subbases generate the same topology

Let $X$ be a set. If $\mathcal B_1$ and $\mathcal B_2$ are bases of subsets of $X$, it is well-known that $\mathcal B_1$ and $\mathcal B_2$ generate the same topology if and only if for any pair of ...
1
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1answer
33 views

a topological property of the product topology

Let $G$ be a non discrete Polish group. Let $K$ be a compact set of $G$, $C$ a closed set of $G^n$ and $B$ an open set of $G^n$. Suppose $K^n\cap C\subseteq B$. Prove that there is an open set of $G$, ...
1
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1answer
24 views

Closed subsets of $\mathbb{C}^*$ proper for multiplication

Let $S_1$ and $S_2$ be two proper closed subsets of $\mathbb{C}^*$. Let's denote by $\overline{S_1}$ and $\overline{S_2}$ their closure in $\mathbb{C}_{\infty}.$ (Alexandrov compactification) ...
0
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4answers
36 views

A sequence in a Hausdorff space and in a space that is not Hausdorff.

Let $X$ be a topological space and $\{x_n\}_{n=1}^{\infty}$ a sequence in $X$. Show that if $X$ is Hausdorff, $x_n \rightarrow x \:$, $x_n \rightarrow y \:$ implies $x=y$. Give an ...
1
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0answers
34 views

The closure of the closed ball is a closed [on hold]

In how many ways you can show that $\overline{\overline{B}(x,r)}=\overline{B}(x,r)$ where $\overline{B}(x,r)=\lbrace y \in \mathbb{R}^n : d_e(x,y) \leq r \rbrace$, and $d_e$ is the euclidean metric ? ...
2
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1answer
18 views

Sequential compactness of smooth functions

Suppose I have a sequence $u_n$ of smooth functions on the $N$-dimensional reals. If $\|D^{\alpha}u_n\|_{\infty} \leq C_{\alpha}$ for all multi-indices $\alpha$, then is it possible to deduce that ...
4
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2answers
45 views

are two metrics with same compact sets topologically equivalent?

are two metrics with same compact sets topologically equivalent ? I think if the cardinal of set is finite then we have one metric that is the discrete metric and every metric on this set is ...
4
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0answers
46 views

Is the sum of infinitely many open sets open?

Let $X$ be a locally convex space (or, in particular, a normed space). Let $(O_n)_{n=1}^\infty$ be an infinite sequence of non-empty open sets in $X$ such that the sum $\displaystyle\sum_{n=1}^\infty ...
5
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3answers
58 views

Show that two topological spaces are not homeomorphic.

Let $X = (-1,1)$ be considered with the Euclidean metric, and $Y = (0, \infty)$ be given the cofinite topology. Prove that $X$ and $Y$ are not homeomorphic. My current thoughts are that a ...
0
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3answers
61 views

Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X

Here X is a Hausdorff Space. I know that singleton sets, {x}, are closed in a Hausdorff space. Although Im not sure if thats how to use the Hausdorff property. Should I investigate $h=F(x)-x$? Can ...
0
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1answer
20 views

Show that $|d(m,n) -d(n,o) | \leq d(m,o)$ for a metric space

Problem Let $(M,d)$ be a metric space. Show that $$|d(m,n) - d(n,o)| \leq d(m,o) \ \forall m,n,o \in M$$ Since $(M,d)$ is a metric space I know it fufills the triangle inequality. So if I ...
2
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0answers
36 views

Is $(\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}$? Where “$\cong$” means homeomorphic.

I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say ...
1
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0answers
15 views

Special case of noetherian space

A topological space  is called Noetherian if it satisfies the descending chain condition for closed subsets. Now let $X $ be a topological space, and there exists a fix natural number $n $ such that ...
0
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0answers
22 views

Another question from Exercise 6d in section 50 in Munkres' textbook in Topology.

I have a question regarding exercise 6d in section 50 from Munkres' Topology textbook: Exercise 6c in section 50 Munkres' Topology textbook. Show that if $N=2m+1$, then $U_\epsilon(C)$ is dense ...
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0answers
20 views

Cantor-Bendixson rank of a first countable space

This question has been bothering me for quite a while, so let me ask it here. Is there a first-countable compact space $X$ with uncountable Cantor-Bendixson index? By a Cantor-Bendixson index I ...
0
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3answers
42 views

Countable connected spaces

I can not think of any countable connected subsets in $\mathbb{R}$ (with subspace topology).. Are there any such? Only countable subsets of $\mathbb{R}$ that i am familiar with is $\mathbb{Q}$ ...
0
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1answer
3 views

Does continuity in one variable and locally Lipschitz in another imply uniformity in the first?

I understand the definition of Lipschitz functions when talking of functions of single variables. However, I have trouble understanding it when it is a multivariable function. Suppose $ f(t,x):D ...
2
votes
1answer
37 views

composition of functions is continuous

Question is as follows : Let $X,Y,Z$ are metric Spaces Let $f:X\rightarrow Y$ be continuous map onto $Y$ and let $X$ be compact. Also $g:Y\rightarrow Z$ such that $g\circ f:X\rightarrow Z$ is ...
0
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1answer
31 views

Convergent Bounded Linear Maps

I'm not sure how to show that the composition of two convergent bounded linear maps converges to the composition of their limits. I've shown that the composition of bounded linear maps is a bounded ...
0
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1answer
24 views

topological invariance of being contained in a set of given dimension

Suppose $U$ is contained in $E^n$ ($n$-dimensional Euclidean space) and is homeomorphic to a set $V$ in $E^m$, where $m>n$. Is there a topological manifold in $E^m$ of dimension $n$ that contains ...
0
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2answers
41 views

Prove that the set $E = \{y ∈ Y : f(y) = g(y)\}$ (a.k.a. the coincidence set of $f$ and $g$) is closed in $Y$

The full question is: Let $X$ be a Hausdorff topological space. (i) Let $Y$ be a topological space and $f, g : Y → X$ be continuous functions. Prove that the set $$E = \{y ∈ Y : f(y) = g(y)\}$$ ...
1
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1answer
40 views

Fréchet derivative of $f(x) = x$

Im not sure how to find the Fréchet derivative of the function $f : \mathbb{X} \to \mathbb{X}$ given by $f(x) = x$, where $\mathbb{X}$ is a normed space. I'm not given the dimension of the normed ...
6
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0answers
80 views

Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where ...
0
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2answers
24 views

Showing the disjoint union topology is a topology

Let $A$ be a set and suppose that for all $\alpha \in A$, we have the topological space $X_\alpha$. Consider the set which is the disjoint union $$ X:=\coprod_{\alpha \in A} X_\alpha. $$ Let $\tau$ be ...
0
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1answer
20 views

An example of the set of distances of two points in two different closed sets having no infimum

On a problem set for my Analysis in Several Dimensions class (basically real analysis on multivariable functions), I encountered this question: Let $(X, d)$ be a metric space, let $C ⊂ X$ be a ...
1
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1answer
27 views

Is every compact metric space hereditarily separable?

Let $X$ be a compact metric space. I see why all open and closed subsets of $X$ are separable. But is every subset of $X$ necessarily separable? EDIT: Since $X$ is separable metric, it embeds into ...
4
votes
1answer
37 views

Does $X^{C_2} \simeq * \simeq X/{C_2}$ imply $X \simeq *$?

What the title says. Let $C_2$ be the cyclic group of order 2, and $X$ be a topological space with a $C_2$-action (acting continuously) such that both the quotient space $X/{C_2}$ and the subspace of ...
2
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2answers
41 views

If $X$ is a set and $\mathcal T$ is the discrete topology on $X$, is the following statement true

If $X$ is a set and $\mathcal T$ is the discrete topology on $X$, is the following statement true: $\{X\} \in \mathcal T$? I know that since $\mathcal T$ is a topology we know that $X ...
1
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1answer
41 views

Is one-point compactification of a space metrizable

Let $X$ be a locally compact Hausdorff space.Let $Y$ be the one-point compactification of $X$. Two questions are: Is it true that if $X$ has a countable basis then $Y$ is metrizable? Is it true ...
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0answers
32 views

Group theory: Intuition as to what a group is [duplicate]

In group theory the group is an algebraic structure consisting of a set which has elements associated with definite finitiary operations. Can an intuitive explanation be provided as to what this ...
0
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0answers
32 views

Why is the unit disc not a topological surface? [duplicate]

I am trying to prove that the unit disc $D^2$ is not a topological manifold. Clearly it is Hausdorff and second countable, so I think I should show that it is not locally Euclidean. The following is ...
0
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0answers
14 views

Ways to prove that two manifolds are $not$ ambient-isotopic to each other

I've just started learning basic topology and have just received an introduction to isotopy, so I apologize if this question appears trivial. What are some of the ways to prove that two manifolds are ...
0
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0answers
22 views

Intersections: Generator

Problem Given a set $\Omega$. Define the generator: $$\mathcal{A}\subseteq\mathcal{P}\Omega:\quad\delta\mathcal{A}:=\{A\cap A':A,A'\in\mathcal{A}\}$$ Then one obtains: ...
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1answer
53 views

What is boundary of $\mathbb{C}$? [on hold]

What is boundary of $\mathbb{C}$? or $\partial \mathbb{C}$?
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3answers
33 views

Show that these metrics induces the same topology on X

Let $X$ be the set of positive integers. Let $d_1$ be the usual metric space on $X$ and $d_2$ be the discrete metric on $X$. Define $d_3:X\times X \rightarrow R$ by ...
61
votes
10answers
3k views

Explain “homotopy” to me

I have been struggling with general topology and now, algebraic topology is simply murder. Some people seem to get on alright, but I am not one of them unfortunately. Please, an answer I need is ...
1
vote
1answer
78 views

Exercise 6c in section 50 Munkres' Topology textbook.

The problem is as follows: Given $f: X \to \mathbb{R}^N$ and given compact subspace $C$ of $X$ ($X$ is locally compact Hausdorff space with a countable basis); let: $$U_\epsilon(C) = \{ f: ...
0
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3answers
53 views

Topological Continuous Functions and Non-Open Sets

Let us consider a function $\ \mathbf F $ defined from $\ \mathbf X $ to $\ \mathbf Y $ , where $\ \mathbf X $ and $\ \mathbf Y $ are topological spaces. Now by definition , a continuous function is ...
0
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0answers
34 views

Simply connectedness of spherical shell

Consider a spherical shell $U$ in $R^3$(the open region between two spheres). I want to show that any closed curve in $U$ can be shrunk into a single point without leaving $U$. This exercise appears ...
0
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0answers
38 views

What does a variable superscript above a set mean?

I'm not entirely sure I've worded this correctly. An example of what I mean is... $$U = \{0,1\}^n$$ What is the meaning of the superscript?
2
votes
1answer
20 views

Show that a connected regular space having more than one point is uncountable

Two questions on which I am stuck: 1.Show that a connected normal space having more than one point is uncountable. 2.Show that a connected regular space having more than one point is uncountable. ...
1
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1answer
29 views

Describing the clopen sets of a profinite group

I've read somewhere that all clopen subsets of a profinite group $$G \simeq \varprojlim\left(G_i, f_{ij}:G_i \to G_j\right)_{i,j \in I}$$ are exactly the preimages of subsets of the $G_i$'s. It's easy ...
2
votes
1answer
32 views

Spaces homotopy equivalent to finite CW complexes

I'm doing a project about Topological Complexity (it doesn't matter what it is for the questions I will ask) and I have proofs for a few results about the bounds of the topological complexity of ...
0
votes
1answer
37 views

Prove closed disc $D^n$ is homeomorphic to the cone $CS^{n-1}$

I need to find a continuous surjective map from $D^n$ to $CS^{n-1}$. For 2 dimensions, we can use $$f: S^1 \times I /S^{1} \times \{1\} \rightarrow D^2$$ with $f(\theta,t) = (1-t)e^{i \theta}$ ...
7
votes
1answer
57 views

Does a map between topologies determine a map between sets?

Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be Hausdorff spaces. Consider a function \begin{equation*} \phi:\mathcal{B}\rightarrow \mathcal{A} \end{equation*} which preserves inclusion, arbitrary ...
0
votes
1answer
33 views

Is torus w. disc removed homotopic to klein bottle w. disc removed?

I know that homeomorhic spaces are homotopic, but am not sure if this applies, since I think they are not homeomorphic due to orientability. I know f and g are homotopic if they represent: ...
1
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1answer
81 views

The magic of the morphisms

Given a set $X$. Let $S\subseteq X$ and consider $(X,S)$ as a very simple mathematical structure, lets call it a spotted set. Given two spotted sets, then a morphism $\alpha ...
0
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1answer
30 views

Differentiability of norm

Is a norm from any normed space $\mathbb{A}$ to $\mathbb{R}$ Frechet differentiable at zero? I've tried proving by contradiction that the norm is not Frechet differentiable at zero, but didn't get ...
0
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1answer
58 views

Showing that $\displaystyle\underset{n\rightarrow \infty}{\lim}\int_0^1 f_n = \int_0^1\underset{n\rightarrow \infty}{\lim} f_n$

How to solve the following task: Show that if $f_n$ is a sequence of uniformly converging mappings $f_n \in C[0,1]$, where $C[0,1]=\{f:[0,1]\rightarrow\mathbb{R} \;\mid\; f\; ...