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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0answers
23 views

compactness of thes sequence set

Let $S$ be a compact (in the usual topology) subset of $\mathbb R^n$, let $W = \{(q_k)_{k\in\mathbb{N}}\,\mid\, q_k\in S\}$ be the set of all the sequences taking elements in $S$, let ...
0
votes
1answer
30 views

Homotopics curves

We are in the plane (x,y). We have two periodic (closed) planar curves : (x1(t),y1(t)) which is a simple loop and (x2(t),y2(t)) which is a limaçon. Are these two curves are homotopic ?
1
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3answers
52 views

Show that the sphere, S, and $\mathbb{R}^2$ is not homeomorphic

I am trying to show that the sphere $S^2$ and $\mathbb{R}^2$ are not homeomorphic.I understand that you can't 'compress' a 3D shape into a 2D plane but I don't know how I would express this formally. ...
1
vote
2answers
54 views

Topology, locally-compact Hausdorff space

I already asked this question here: locally-compact Hausdorff space, equivalent, compact, continuous So if this repost is not apprechiated, please just delete this thread, but I would really like to ...
-3
votes
0answers
48 views

Prove that $[0,1]$ is compact. [on hold]

Prove that $[0,1]$ is compact. Using the definition of compactness. (Not using Heine-Borel theorem)
0
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1answer
27 views

Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$

I have the following exercise: Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$ I don't know what $b$ is meant to be, there's a typo in this exercise. I ...
2
votes
2answers
35 views
+50

Proof that a discrete space (with more than 1 element) is not connected

I'm reading this proof that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton ${x}$ and its ...
0
votes
1answer
21 views

$A$ is an open subset of $M$ $\iff$ ($x_n\to a\implies x_n\in A$ for large $n$)

My definition of an open subset $A$ of $M$ is the one that for every $x\in A$, there is an open ball contained in $A$. Now, suppose that $x_n\to a$. By definition, $\forall \epsilon>0$ there exists ...
3
votes
1answer
42 views

Show completeness of metric subspace

I have problems solving the following 2 problems: Given is the metric $d:\Bbb R\times\Bbb R\to[0,\infty[$ with $$d(x,y):=|\arctan(x)-\arctan(y)|\;.$$ a) Show that the metric subspace ...
0
votes
0answers
23 views

Questions about proof of $\lim x_n = a, \lim y_n = b\implies \lim x_n+y_n = a+b$ in a normed vector space

I need to prove that, in a normed vector space $E$, we have: $$\lim x_n = a, \lim y_n = b\implies \lim (x_n+y_n) = a+b$$ and: $$\lim\lambda_n = \lambda, \lim x_n = a \implies \lim \lambda_n\cdot ...
4
votes
2answers
39 views

Homotopic retract vs deformation retract

Let's say that $A \subset X$ is a deformation retract. It follows that $A$ is both a retract and a space homotopically equivalent to $X$. Is the converse true? Probably not, but I couldn't find any ...
0
votes
1answer
26 views

Exercise I.7.2 in Geometry and Topology by Bredon

I'm working though the first chapter in Geometry and Topology by Glenn Bredon, and I'm stuck on Exercise I.7.2, which is related to compactness. It reads: Let $X$ be a compact space and let ...
2
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1answer
36 views
0
votes
1answer
29 views

Showing $\mathbb{B}_{\mathbb{Q}}$ is a bases for $\mathbb{R}_{\text{usual}}$

Show that the collection $\mathbb{B}_{\mathbb{Q}} := \{(p, q) \subseteq \mathbb{R} : p, q \in \mathbb{Q}, p < q \}$ is a basis for the usual topology on $\mathbb{R}$. Solution: We know that ...
5
votes
0answers
32 views

Regularly open, co-zero sets in compact Hausdorff spaces

It follows from the definition of a completely regular space that such spaces have a base consisting of co-zero sets, that is, sets whose complement is the zero set of some real-valued, continuous ...
39
votes
5answers
2k views

What is a topological space good for?

I know there are already some questions similar to this, which all give an answer that a topological space creates some structure on a set which is an abstraction of distance and makes it possible to ...
0
votes
0answers
27 views

Deriving a bounding $\delta$ of an interior point

This question is based on the Baby Rudin's 2.16: Regard $Q$, the set of all rational numbers, as a metric space, with $d(p,q)=\lvert p -q \rvert$. Let $E$ be the set of all $p \in Q$ such that $2 ...
1
vote
2answers
29 views

If two sequences are Cauchy, then d(sequence_1, sequence_2) is cauchy in R

The question says this: If $(X,d)$ is a metric space and $\{x_n\}$ and $\{y_n\}$ are Cauchy sequences, prove that $\{d(x_n,y_n)\}$ is a Cauchy sequence in $R$. I see that I would have to show that ...
-2
votes
1answer
46 views

Do compact connected smooth manifolds admit the structure of a CW complex with a single 1-cell? [on hold]

This seems intuitive to me, since they admit a CW decomposition with finitely many cells. But I can't see how to prove it.
1
vote
1answer
36 views

Is the distance function open?

I know that any distance function is continuous under the topology induced by it. Does it also have to be an open mapping?
0
votes
1answer
13 views

Is the weak-* topology on a topological vector space Hausdorff?

Let $V$ be a topological vector space and $V^*$ be the space of linear functionals induced with the weak-* topology. Can we say that $V^*$ is Hausdorff? Here is my attempt: Let $\lambda\ne\lambda'\in ...
1
vote
1answer
21 views

Identifying the antipodal points of one boundary of the cylinder gives the Möbius band.

In Example 1.35, Hatcher writes in his book Algebraic Topology, the following (not paraphrased): Let $X=S^1\times I$, and let $A$ be the quotient space obtained by defining the relation $(z, ...
0
votes
1answer
22 views

How to find an open set $W$ around $1\in S^1$ so that $\Delta\subset \{(x, wx)\mid x\in S^1, w\in W\} \subset U$ for an open $U$?

This is a question on a practice topology qual. Here is the full wording: Let $U$ be an open subset of $S^1\times S^1$ containing $\Delta = \{(x, x)\mid x\in S^1\}$. Show that there exists an open ...
0
votes
1answer
22 views

Show that [0, 1) with the induced topology from R is a Polish space.

It's easy to see that the space is separable because $Q \cap [a,b)$ is a countably dense subset of $[a,b)$, but I can't figure out a way to show that it's completely metrizable. I know this means ...
0
votes
0answers
44 views

Topology bases for $\mathbb{R}_{\text{usual}}$

I'm trying to compile correctly formulated solutions to common topology questions as a summer project. I'm not very confident in my proof writing abilities so I'm going to post my solutions here for ...
0
votes
1answer
33 views

In the co-finite topology and the co-countable topology, must $X$ be finite or countable?

Recall $\tau_{co-finite} = \{U \subseteq X| X\backslash U \text{ is finite}\}\cup\{\varnothing\}$ $\tau_{co-countable} = \{U \subseteq X| X\backslash U \text{ is countable}\}\cup\{\varnothing\}$ ...
0
votes
1answer
35 views

Hausdorff compact problem

Let $X$ a Tychonoff space and the topological immersion $e: X \to \prod_{s \in S} [0,1]$. For this other question: Show that for all compact $K$ and for all continuous function $f:X \to K$, there is ...
1
vote
3answers
21 views

Given $A \subseteq X$ in the discrete and the trivial topology, find closure of $A$

Given $A \subseteq X$ in the discrete and the trivial topology, find closure of $A$ Note the definition of closure I am using is one in Munkres: $x \in \overline A \iff \text{ for every ...
1
vote
2answers
35 views

Questions about Proof that Cartesian Product of Open Sets is an Open Subset

I'm trying to understand the proof that: The cartesian product $A_1\times \cdots\times A_n$ of open subsets $A_i\subset M_i$ is an open subset of $M=M_1\times\cdots\times M_n$. It follows ...
1
vote
2answers
44 views

Does the Hausdorff property hold on closed subsets of $\mathbb{R}^n?$

I am trying to prove that given disjoint closed $A,B\subseteq \mathbb{R}^n$, there exist disjoint open $U,V$ containing $A,B$ respectively. In other words that we can take the Hausdorff property to ...
1
vote
1answer
20 views

Trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$

I am trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$ I know that $RP^2$#$T^2$~$RP^2$#$K^2$ and that $X(M$#$N)$=$X(M)+X(N) -2$ where X is the ...
1
vote
1answer
39 views

Simple example of a mapping between topological spaces

I read the definition of a continuous function between topological spaces a lot of times, but I'm having difficulties to apply it to a simple example. Given two topological spaces $(X,\tau_1)$ and ...
1
vote
2answers
40 views

Constructing topology on $\Bbb{Z}$

Fix an infinite subset $A$ of $\mathbb Z$ whose complement $\mathbb{Z}\setminus A$ is also infinite. Construct a topology on $\mathbb{Z}$ in which: (a) $A$ is open (b) Singletons are never open (i.e ...
1
vote
2answers
31 views

analogue of the Jordan curve theorem for closed curve

I wonder whether there are some generalization of the Jordan curve theorem : Can the theorem be generalized into closed curve? $C$ is a closed curve , then $\Bbb R^2\setminus C$ consists of several ...
0
votes
2answers
37 views

homeomorphism from interval $[a,b]$ to $[0,1]\subset \mathbb{R}$

I need to show that every interval $[a,b]$ is homeomorph to $[0,1]\subset \mathbb{R}$. I've found this answer but it only deals with open sets, and I need an answer that deals with closed sets.
1
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0answers
44 views

What to do after defining a metric on a set? [on hold]

Given a finite set $M$ of binary sequences of length 6: $$ M=\{\{1,0,1,0,0,1\},\{1,0,0,0,1,1\},...\} $$ Let's define a metric (Levenshtein distance) on $M$, which makes it a metric space. That's ...
0
votes
0answers
16 views

Continuous indicator-like functions

Let $\Omega$ be a compact subset of $\mathbb{R}^n$. Let $g:x\in\mathbb{R}^n\to\mathbb{R}$ be a continuously differentiable function such that $$ \begin{cases} g(x)>0 & x\in\text{int}\Omega,\\ ...
0
votes
2answers
19 views

Sequence of partial sums of e in Q is a Cauchy sequence.

Verify that $X_n= \{ \sum_{i=0}^n$ $\frac{1}{i!}$} is a Cauchy sequence in $Q$ with the Euclidean metric. I can't figure out how to find an $N$ that makes this work. I figure that $d(x_n,x_m) < ...
2
votes
1answer
45 views

morphism of sheaves on $\mathbb{R}/\mathbb{Z}$

Let $\mathscr{Z}$ be an arbitrary sheaf on $\mathbb{R}/\mathbb{Z}=X$ (with the quotient topology). Let $\mathscr{F}$ and $\mathscr{G}$ denote the sheaves of continuous functions on $X$ with values in ...
2
votes
1answer
28 views

How is a solid doughnut, a disk of circles?

I understand how its a circle of disks but not how its a disc of circles. Solid doughnut = $D^2$ X $S^1$. Animation would elate me.
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0answers
24 views

Stone-Cech compactification and maximal ideal of C(X)

Please i want to know the usefulness of the "Gelfand and Kolmogoroff theorem" when showing that $\beta X$ of a a completely regular Hausdorff space $X$ Can be identified with the Structure space of ...
0
votes
1answer
29 views

Induced homomorphism from homology group of circle to homology group of $\mathbb{R^2-}0$ is trivial

Let $C_r$ be a circle of radius $r$ in complex plane, and let $f:C_r\to\mathbb{R^2}-0$ defined by $f(z)=z^n+a_{n-1}z^{n-1}+...+a_0$ and suppose that it has no zero on and inside the circle $C_r$. ...
8
votes
2answers
367 views

The reason behind the definition of manifold

I was going thorough the definition of a manifold and needless to say it wasn't something that I could digest at one go. Then I saw the following Quora link and Qiaochu's illustrative answer. It was ...
0
votes
0answers
16 views

For manifolds $M,N$ show that $W^{1,p}(M,N)$ is path-connected iff $C^0(M,N)$ ist path-connected.

I'm asked to show that for compact, smooth Riemmanian manifolds $M,N$ we have that $W^{1,p}(M,N)$ is path-connected if and only if $C^0(M,N)$ is path-connected. The theorem (0.1) is taken from ...
0
votes
1answer
54 views

Borel set as union of $G_\delta$ and countable set

What is an example of a Borel set of $\mathbb{R}$ which cannot be written as a union of a $G_\delta$ and a countable set?
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0answers
24 views

Proof that addition on a Banach space is continuous

What I have so far: Let $(W,+,\cdot,\Vert\cdot\Vert)$ be a Banach space. We have the map $$+ : W\times W \to W.$$ The topology $T$ on $W$ is given by $$T = \{U\subseteq W ~\big|~ \forall u\in U : ...
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vote
2answers
60 views

A Natural Question When Reading Van Kampen Theorem

Let $A$ and $B$ be path connected open subspaces of a topological space $X$ and assume that $A\cap B\neq \emptyset$ is simply-connected. Let $x$ and $y$ be two points in $A\cap B$. Let $\gamma$ and ...
2
votes
1answer
40 views

What does $X_j \approx X$ mean when used in this blog post?

I was trying to learn disjoint union topology and used the following blog : https://drexel28.wordpress.com/2010/04/02/disjoint-union-topology/ The second theorem about disjoint topology says that if ...
0
votes
1answer
24 views

A continuous action of a compact group on a uniform space is equicontinuous?

I am stuck on how to prove the statement "A continuous action of a compact group $G$ on a uniform space $X$ is equicontinuous." So, essentially we want to show that for every entourage $\alpha$ of ...
1
vote
0answers
24 views

Rewording the definition of closure

In Munkres there was a statement: Given a topological space $(X, \tau)$ $x \in \overline A \iff \text{ for every open set } U \text{ containing } x, U \cap A \neq \varnothing$ Following from ...