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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How do I know the fundamental group of an infinite graph is well defined?

I get that given a choice of spanning tree and base point for a (connected) graph, I can effectively change the base point through path conjugation, so there's no problem there. For finite graphs, the ...
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1answer
37 views

Definition of a separable metric space

The book I'm reading doesn't explicitly give a definition of separable metric spaces. The only type of separability definition I know that a separable topological space is one that has a countable ...
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2answers
30 views

Proving equivalence of statements on continuity between metric spaces

On page 228 of Mícheál Ó Searcóid's Metric Spaces (2007), he writes Criteria for Comparability of Metrics Suppose $X$ is a set and $d$ and $e$ are metrics on $X$. Then the following ...
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0answers
22 views

Continuously variable *space* [on hold]

I'm trying to understand formally how and why a fiber bundle with fiber $F$ should be thought of as a gluing of homeomorphic copies of $F$ which varies continuously. I do not understand how this is ...
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1answer
28 views

Is the Kähler differential of a continuous function ring trivial?

Suppose $A=C^0(\mathbb R)$ is the ring of real-valued continuous functions on $\mathbb R$. Is it true that, the Kähler differential $\Omega_{A/\mathbb R}$ trivial? In other words, suppose that ...
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37 views

How does the compactness property help us show a subset $A$ of a metric space $X$ is closed?

We have a compact subset $A$ of a metric space $X$ and we want to show that this implies that $A$ is closed. Let $y \in A$ and $y \in A^c$. For each $y \in A$, we can take open neighbourhoods $U_y$ ...
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37 views

Prove $\bar{A}\setminus \bar{B} \subset \overline{A\setminus B }$

Here is my approach so far Let $A$ and $B$ be subsets of the metric space $(M,d)$ My thoughts on how to prove it, is to choose an element $x$ from $\bar{A}\setminus \bar{B}$ and show it exists in ...
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1answer
44 views

Set having a enumerable dense subset [closed]

I've got an exercise in which I need to prove the following: A set has an enumerable dense subset. Can you share the ways to do this? Edit: X a normed vector space $Y:=B(X,\mathbb{K})$ a vector ...
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1answer
30 views

Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a ...
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1answer
25 views

Constructing a metric topology that is the same as the standard topology

Any help on this problem would be greatly appreciated. thanks! $\textbf{Definition:}$ Let $\tau$ be the collection of subsets of $\mathbb{R}^n$ with the following property: $\forall x \in U,\; ...
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1answer
26 views

Proving that sums of convergent sequences are complete metric spaces

Let $L_1$ be the set of all sequences of real numbers $$x = (x_1,x_2,..., x_n, ...) $$ with the property that $\sum_{n=1}^\infty |x_n|$ is convergent. If we define $$d_1(x,y) = \sum_{n=1}^\infty ...
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41 views

Proof: $X$ is Hausdorff if and only if the diagonal $\Delta$ is closed in $X\times X$.

Prove that $X$ is Hausdorff if and only if the diagonal $\Delta$ is closed in $X\times X$. This exercise appeared on a previous exam in my course, and also in Munkres. Here's my attempt: First I ...
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0answers
24 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 ...
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1answer
31 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 ?
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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. ...
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2answers
55 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 ...
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0answers
48 views

Prove that $[0,1]$ is compact. [closed]

Prove that $[0,1]$ is compact. Using the definition of compactness. (Not using Heine-Borel theorem)
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1answer
29 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 ...
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3answers
57 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 ...
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1answer
22 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 ...
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1answer
48 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 ...
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0answers
25 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 ...
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2answers
40 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 ...
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1answer
27 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 ...
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1answer
36 views
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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 ...
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0answers
36 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 ...
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5answers
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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 ...
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1answer
40 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 ...
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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 ...
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1answer
48 views

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

This seems intuitive to me, since they admit a CW decomposition with finitely many cells. But I can't see how to prove it.
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1answer
37 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?
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1answer
14 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 ...
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1answer
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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, ...
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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 ...
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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 ...
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0answers
45 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 ...
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1answer
33 views

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

Recall $\tau_{co-finite} = \{U \subseteq X| X\backslash U \text{ is finite}\}\cup\{\varnothing\}$ $\tau_{co-countable} = \{U \subseteq X| X\backslash U \text{ is countable}\}\cup\{\varnothing\}$ ...
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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 ...
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3answers
23 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 ...
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2answers
36 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 ...
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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 ...
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1answer
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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 ...
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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 ...
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2answers
41 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 ...
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2answers
32 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 ...
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38 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.
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0answers
44 views

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

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 ...
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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,\\ ...
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2answers
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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) < ...