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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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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13 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 ...
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3answers
36 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 $\mathbb{Q}$ ...
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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 ...
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1answer
25 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 ...
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1answer
20 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 ...
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2answers
30 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 $$E = \{y ∈ Y : f(y) = g(y)\}$$ ...
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1answer
35 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 ...
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1answer
69 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 ...
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2answers
23 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 ...
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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 ...
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1answer
26 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 ...
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0answers
35 views

Hopkins Algebraic Topology notes, something not clear involving Stiefel-Whitney classes.

In notes by Mike Hopkins here, the following is remarked. A discussion similar to the one for $\textbf{RP}^n$ shows that$$w(T\textbf{CP}^n) = (1 + y)^{n+1}$$where $y \in H^2(\textbf{CP}^n, ...
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1answer
35 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 ...
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2answers
40 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 ...
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1answer
39 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 ...
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0answers
31 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 ...
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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 ...
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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
51 views

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

What is boundary of $\mathbb{C}$? or $\partial \mathbb{C}$?
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3answers
31 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 ...
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10answers
2k 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 ...
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1answer
76 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: ...
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3answers
52 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 ...
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0answers
33 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 ...
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0answers
37 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?
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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. ...
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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
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1answer
28 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 ...
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1answer
36 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 $f(\theta,t) = (1-t)e^{i \theta}$ ...
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47 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 ...
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1answer
32 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: ...
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1answer
79 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 ...
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1answer
28 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\; ...
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0answers
21 views

any sum of sets open\nullset is a set of the same form

I'm curious how can one prove that any sum of sets $G\setminus N$, where $G$ is open and the Lebesgue measure of $N$ is 0, is a set of the same form. it is easy for countable sums, but in general? ...
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0answers
38 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
0
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2answers
32 views

Conditions for a homeomorphism

In my topology notes the definition is given as: A function $f : X \rightarrow Y$ is said to be a homeomorphism if: $f$ is continuous, bijective, and moreover its inverse $f^{−1} : Y \rightarrow X$ ...
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0answers
29 views

how can i prove that 3 problems [on hold]

Prove that any subspace of a discrete space is discrete. Prove that any subspace of an indiscrete space is indiscrete. Prove that if A C X is r-open, then any r A -open set is also r-open.
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1answer
34 views

Fundamental Property of Regular CW Complexes

$\newcommand{\R}{\mathbf R}$ For a cell $e$ in a CW complex, we write $\partial e$ to denote $\bar e-e$. Note that $\partial e$ may not be the topological boundary of $e$ in $X$. A CW complex ...
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1answer
22 views

Equivalent properties for a kind of relative compactness?

Let $X$ be a topological space and $Y \subseteq X$. Consider the following statements: (i) Every net in $Y$ has a cluster point in $X$. (ii) Every infinite subset of $Y$ has a complete accumulation ...
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1answer
17 views

Hybrid equivalence of Polynomial-like maps

I am reading Douady and Hubbards "On the dynamics of polynomial-like mappings". I am relatively new dynamics of complex maps, and I would appreciate some help with aspects of the following. ...
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1answer
19 views

Should a compact nbd of $p$ contain some open nbd containing $p$?

If some topological space $X$ is locally compact, then each point of $X$ has at least one compact neighborhood. The book I'm reading now doesn't mention if such compact neighborhood of a point should ...
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1answer
36 views

Proving $\mathbb{R}/\sim$ is homeomorphic to unit circle

Let $S$ be the unit circle in $\mathbb{C}$, standard topology. Define the equiv. rel. $\sim$ on $\mathbb{R}$ as $x\sim y\iff x - y\in\mathbb{Z}$. I would like to prove that $\mathbb{R}/\sim$ is ...
0
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1answer
19 views

The degree of a circle function

Does the degree of a circle function $f:S^1 \to S^1$ simply mean how many times the mapping of $f$ wraps around the $S^1$?
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2answers
15 views

Prove that the space $\Bbb R_K$ is not regular.

Prove that the space $\Bbb R_K$ is not regular. where the basic open sets on $\Bbb R_K$ is given by $\{(a,b):a,b\in \Bbb R\}\cup \{(a,b)-K\}$ where $K=\{\dfrac{1}{n}:n\in \Bbb Z_+\}$. ...
2
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1answer
54 views

If $x$ is an isolated point of $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$. [duplicate]

Is the following proof valid? (Note: I know there is a post discussing this problem, but I am curious to see if my argument works). This problem is different from another post that is similar with ...
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0answers
36 views

Let $S$ be the set . Which of following are true?

Let $S =\{\frac{1}{3^m}+\frac{1}{7^n}$ , where $m,n \in \mathbb N\}$ Then A.$S$ is closed B.$S$ is not open C.$S$ is connected D.$0$ is a limit point of S I see that $0$ is limit point of $S$ but ...
4
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1answer
50 views

What are the two disjoint closed sets that cannot be separated by two disjoint open neighborhoods in the Ellentuck topology?

Denote by $X := [\mathbb{N}]^\infty$ the set of infinite subsets of $\mathbb{N}$. Recall that the Ellentuck topology is a topology on $X$ generated by sets of the form $\{A\text{ infinite} \mid ...