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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1answer
17 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 ...
5
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0answers
26 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
17 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
18 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
25 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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31 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, ...
4
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1answer
34 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
38 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
37 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
29 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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13 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
21 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
49 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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9answers
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
69 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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31 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
36 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
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1answer
19 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
26 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
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1answer
33 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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46 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
76 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
27 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 ...
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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
37 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 ...
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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.
1
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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 ...
1
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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. ...
2
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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 ...
0
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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 $S^1$?
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
53 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 ...
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1answer
25 views

Prove some identities involving interior, closure in topology

I want to prove the following statements: (i) $X \setminus Y ^ \circ $ = $cl ( X \setminus Y) $ I wrote down that $Y ^\circ$ is open so $Y ^\circ = Y $. Therefore $X \setminus Y ^ \circ $ is ...
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1answer
60 views

What kinds of transformations preserve network topology?

I have been reading a number of "network science" papers where the authors perform transformations on networks that seem to preserve the topology of those networks. By "topology", I mean a collection ...
1
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1answer
21 views

Connected sets definition [duplicate]

Defn: A set $X$ is connected if there do not exist non-empty, disjoint open sets $U,V$ s.t $U$ $\cup$ $V$ $=X$. I thought intuitively that this meant that this was like the English dictionary ...
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1answer
24 views

Compact sets in the product of topological spaces.

Let $G_1$ be a non-compact topological space and let $G_2$ be a generic topological space. What are the compact sets in the product $G_1\times G_2$? Surely we can take the sets of the form $K_1\times ...
1
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1answer
38 views

The point-wise closure of the space of continuous functions

Let $X$ be a locally compact space and consider $C_0(X)$. We denote $b(X)$, by the set of all bounded functions on $X$. It is easy to be checked that $b(X)$ may be considered as a C*-sub algebra of ...
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2answers
32 views

Interior, closure and boundary of subset in $\mathbb{R}$ \ $\mathbb{Q}$.

I have the subset $\left[0,1\right] \backslash \mathbb{Q}$ in $\mathbb{R} \backslash \mathbb{Q}$. Am I right in thinking that this set is open and not closed in the space given? Also, how do I go ...
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1answer
74 views

Injectivity in the zero homology

I'm struggling with following step in an excercises about Mayer-Vietoris sequences: In one step the solution says this map is injective since $A \cap B$ is path-connected: $$ H_0(A \cap B) ...