Tagged Questions

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
80 views

Show that a set $S$ is closed if and only if $S=\operatorname{int}(S)\cup \operatorname{Boundary}(S)$

Show that a set $S$ is closed if and only if$S=\operatorname{int}(S)\cup \operatorname{Boundary}(S)$. I´m using the following definitions: A set is closed if it is the complement of an open set. ...
0
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0answers
40 views

A homeomorphism preserves irreducible components?

Let $f$ a homeomorphism between two Hausdorff topological spaces $X$ and $Y$. Assume that $X$ and $Y$ are reduced analytic spaces. Is true that $f$ takes an irreducible component of $X$ in an ...
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1answer
29 views

Understand the definition of a neighborhood of a point in a topological space

I'm a physics grad student and recently I decided that I would properly learn differential geometry. I then decided that I would start to learn what a topological space is then build up from there ...
1
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1answer
47 views

Question about space filling curves

I was recently taught that the Peano curve is an example of a continuous bijection from the closed unit interval to the closed unit square. However, if we take a point in the square and take it's ...
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2answers
33 views

Understanding a lemma regarding subspace topology

I am learning topology and am struggling with some of the concepts of open sets regarding subspaces. The problem I am working on says, that Y is a subspace of X, and A is a subset of Y, then show ...
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0answers
30 views

differences between $Y^X$ and $C(X,Y)$ in topology

Can someone please explain me what are the definitions and differences between $Y^X$ and $C(X,Y)$ in topological spaces?
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1answer
26 views

Completion and Seprability of C[0$\infty$]

If I have C[0,$\infty$] the space of all continuous functions on [0,$\infty$] with metric $$ \phi(\omega_1, \omega_2) = \Sigma^{\infty}_{n=1} (1/2^n)*max_{0{\leq} t {\leq} ...
1
vote
1answer
26 views

Finite intersection property in any metric space

If $(X,d)$ is any metric space and $\{A_\alpha\}_{\alpha\in I}$ is a collection of nonempty compact subsets of $X$ such that the intersection of any finite subcollection of sets is non empty does that ...
0
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1answer
24 views

inverse of stiefel-whitney class of product bundles

Let $\xi_1,\cdots,\xi_n$ be vector bundles over $M$. Let $\omega$ be the Stiefel-Whitney class and $\bar\omega$ be the inverse. By an exercise in Milnor's book, $$ w(\Pi_{j=1}^a ...
2
votes
1answer
79 views

Every compact set in $\mathbb{Q}$ has empty interior proof

Consider $\mathbb Q$ with the subspace topology. I read on planet math website that every compact subset in this space has empty interior and then I tried to prove it. Please could someone tell me if ...
0
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1answer
24 views

metric induced norm or norm induced metric

I am always confused between whether metric induces norm or norm induces metric. Can someone clarify this relation a bit for me? Was there any intuition on which direction being true?
0
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1answer
38 views

Is it true that $\textrm{supp}(f)\subseteq K$ implies $f|_{\partial K}=0$?

Maybe this will be an elementary question but I need to clarify this. Let $X$ be a metric space and let $f:X\longrightarrow \mathbb R$ continuous. Suppose $\textrm{supp}(f)\subseteq K$ where $K$ is ...
2
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1answer
23 views

Counterexample on weaker version of result about compact sets

The following is a very well known theorem: Let X be a metric space. $K \subset X$ is compact iff every collection $ \{ F_j \}_{j\in A}$ of closed sets with the finite intersection property in K ...
2
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1answer
44 views

A condition for a covering map to be regular

We said that a path-connected covering map $p:E \rightarrow X$ is regular if: $\forall e \in p^{-1}(x_0): p_{\sharp} \pi_1(E,e)$ is a norm subgroup of $\pi_1(X,x_0)$. or equivalently: If closed ...
0
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1answer
48 views

Gluing oriented manifold along boundaries

Let $M_1$ and $M_2$ be oriented manifolds with boundaries. Suppose they have homeomorphic boundaries. I want to glue $M_1$ and $M_2$ along the boundaries via some homeomorphism. To ensure that the ...
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2answers
76 views

A perfect Hausdorff space that is not metrizable.

Can anyone provide an example of a well-known topological space that has the following three properties: (1) It is perfect (contains no isolated points), (2) T2, and (3) not metrizable.
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1answer
29 views

Integral closure as topological closure

For a commutative ring $A$ you can define the integral closure of $A$ as $$\overline{A}^{\operatorname{int}}:=\lbrace x\in \operatorname{Quot}(A)\mid x\text{ is integral over } A \rbrace.$$ Since this ...
0
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1answer
39 views

Normal spaces that are not necessarily T1

Let $X$ be a topological space. $X$ is normal provided any two disjoint closed subsets may be separated by disjoint open subsets. $X$ is hereditarily normal provided every subspace is normal in the ...
1
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1answer
52 views

Closure of the product of two sets vs product of their closures.

This exercise is from Chapter 2, Section 17, number 9, pag. 101, from Munkres's Topology. Let $A \subset X$ and $B \subset Y$. Show that in the space $X \times Y$, we have $\overline {A \times B} = ...
3
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2answers
68 views

Urysohn's lemma with Lipschitz functions

In a complete and separable metric space $(X,\mathrm{d})$ given an open set $U$ and a closed set $K\subset U$. Is it possible to find a Lipschitz function $f$ such that $f|_K=1$ and $f|_{X\setminus ...
0
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0answers
28 views

About the conditions in Jordan's Curve Theorem

In the original formulation of the theorem, it was stated that a Jordan curve separate the plane in two sets that is not path-connected. The formulation in Wikipedia is that the Jordan curve separate ...
4
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1answer
73 views

Fundamental group of the complement of an eight shape

What is $\pi_1(\mathbb{R}^3 \setminus (S^1 \vee S^1))$? I would say that the space deformation retracts to a sphere $S^2$ surrounding the missing eight with two sticks stuck in the two loops of the ...
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2answers
24 views

Why does the second requirement of a Basis for a Topology make sense?

Why does the second requirement of a basis, i.e. that a point in two sets must also be contained in a third set contained in the intersection of the two set, make sense? Why is that what we should ...
0
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0answers
52 views

How to prove a rectangle is compact

Show that the rectangle $K=[0,a]\times [0,b]$ is a compact subset of $\mathbb{R}^2$. My try : Take some open cover $\{U_{\alpha}\}_{\alpha\in \Lambda}$ of $K$. Now I tried to prove it is closed and ...
3
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1answer
30 views

“Heine–Borel” for the Sorgenfrey line [duplicate]

The Heine–Borel theorem perfectly characterizes the compact subsets of the real line $\mathbb{R}$ (with the usual metric/order topology): Heine–Borel Theorem. A subset $A \subseteq \mathbb R$ is ...
-1
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3answers
30 views

Let $O_1$ and $O_2$ be open sets in $R$, prove that $O_1 \times O_2$ is an open set in $R^2$

How would you prove the following? If $O_1$ and $O_2$ are open sets in $R$, then the set given by $O_1\times O_2$ is an open set in $R^2$ I'm understanding open as: $O$ is open if every point of ...
0
votes
1answer
27 views

Characterizing the family of Borel subsets of a subspace

Given any topological space $X$, let $\mathcal{B}(X)$ denote the $\sigma$-algebra of Borel subsets of $X$. Let $X$ be a topological space and let $Y\subset X$ be given the relative topology. Then ...
1
vote
1answer
50 views

Separable implies second countable

We have $(X,d)$ a metric space. The problem I want to prove is quite long so I'll just put what I need to get it: if $X$ is compact then is separable if $X$ is separable then is second countable ...
0
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2answers
29 views

Showing that we can represent a finite union of intervals as a finite union of pairwise disjoint intervals

Let $K$ be a family of subsets of $\mathbb{R}$ s.t. $A \in K$ iff $A$ can be represented as a finite union of intervals of the form $[x,y)$. Show that $\forall A \in K$ can be represented as a finite ...
1
vote
1answer
68 views

general topology (self learning)

Hi everyone I'd like to know if the following is correct. I'd appreciate any suggestion. Thanks in advance. From Dudley´s book: Let $A_n$ be the set of all the integers greater than $n$. Let ...
0
votes
1answer
39 views

Does continuity follow from linearity on all or only finite-dimensional vector spaces

I'm currently reading an introduction book on topology. While solving one of its exercises I came across something odd. The exercise is: Let $E$ and $F$ be normed spaces, let $T:E \to F$ be linear, ...
1
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0answers
32 views

Is there a non-empty subset of $\mathbb{R} $ like $A$ such that the set of accumulation points of $A$ is itself and $A\cap\mathbb{Q} = \emptyset $ [duplicate]

Is there a non-empty subset of $\mathbb{R} $ like $A$ such that the set of accumulation points of $A$ is $A$ and $A\cap\mathbb{Q}=\emptyset\,$?
0
votes
1answer
51 views

Is the pullback of two covering spaces $\tilde X$ and $\hat X$ a covering space?

Suppose we have two covering spaces $p:\tilde X \rightarrow X$ and $q:\hat X \rightarrow X$ of the same space. Is the pullback $\tilde X \times_X \hat X$ also a covering space of $X$? If yes, what ...
0
votes
2answers
81 views

Is there a continuous surjection from the closed unit square $[0,1]\times[0,1]$ to $\mathbb R ^2$?

Is there a continuous surjection from the closed unit square $[0,1]\times[0,1]$ to $\mathbb R ^2$? If yes, please give examples. I'm a little stuck on this. What if I replace the closed unit ...
3
votes
0answers
30 views

Compactness in minimax theorem

According to Von Neuman's minimax theorem we have $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$ for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) ...
0
votes
3answers
43 views

A map $f:([a,b], |\cdot|) \to ([c,d], |\cdot|)$ is an isometry if and only if $d-c = b-a$.

I was asked to prove the following problem: A map $f:([a,b], |\cdot|) \to ([c,d], |\cdot|)$ is an isometry if and only if $d-c = b-a$. But I think this is not correct, specifically the sufficient ...
0
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0answers
33 views

Is the number of open sets same in homeomorphic topological spaces?

Does the homeomorphic topological spaces, have same number of open sets?
0
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1answer
38 views

How to prove the equivalence of 2 affine spaces given that one is the subset of the other one?

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
1
vote
1answer
55 views

open sets in the order topological space

I have a question. I am really confused about determining if a set is open. First, the idea of a set being closed has nothing to do with homomorphic ideas of closure: if $x,y \in F$ then $x+ y$ ...
2
votes
0answers
75 views

Introductory Topology True/False check - Topology without tears - Exercises 1.1

I have just started to learn Topology, using specifically the book mentioned in the title. I have placed that information in the title with SEO in mind, if this is not acceptable practice in this ...
0
votes
1answer
49 views

strictly finer topologies and bases

i have just a quick question, if T1 > T2 where T1 is strictly greater than T2 are their respective bases strictly greater? where T_i are topologies on a set X . i dont think so because since ...
3
votes
1answer
88 views

Questions about simplex and affine space

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
0
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0answers
34 views

The closure of $\{x\times0\mid 0<x<1\}$

I just want to know if I am right The closure of $\{x\times0\mid 0<x<1\}$ is $\{[0,1]\times0\}$ with the ordered square topology, is that right?
0
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1answer
44 views

triangle inequality for a metric space

If $d_{\infty}(a,b) =$ max$\{|a_{i} - b_{i}|\}$ for $1 \leq i \leq k$, I want to prove that this is a metric on $\mathbb{R}^k$. Its pretty clear that $d_{\infty}(a,a) = 0$ and it is also pretty clear ...
4
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2answers
63 views

Prove that the set $\{(x,y) \in\mathbb R^2\mid y<x^2\}$ is an open set by giving an explicit radius

I´m basically trying to prove that if $S$=$\{(x,y) \in\mathbb R^2\mid y<x^2\}$. Then, for any $(x,y) \in S$ there exists a radius $\delta$ such that $B_\delta(x,y) \subseteq S$. What values of ...
1
vote
1answer
54 views

$S^{n-1}$ is not a deformation retract of $\mathbb{P}^n(\mathbb{R})/ B(0,1)$.

Let $n$ be $\geq 2$, $$\mathbb{P}^n(\mathbb{R}) \supset S^{n-1}= \lbrace [1,x_1,...,x_n] | x_1^2+...+x_n^2=1 \rbrace$$ and $B(0,1)= \lbrace [1,x_1,...,x_n] | x_1^2+...x_n^2<1\rbrace $. Show that ...
0
votes
2answers
46 views

Name for “bicontinuous function” that's not bijective?

We know that an invertible continuous function whose inverse is also continuous is called a homeomorphism. But is there a name for a not-necessarily-bijective function that is "bicontinuous" in the ...
1
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4answers
193 views

A Euclidean space that is homeomorphic to a non-Euclidean space

Is there a well-known example (preferably in dimensions 2 or higher) of two homeomorphic spaces: (1) a metric space with the Euclidean metric and (2) a metric space that is not Euclidean?
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2answers
90 views

Good book on Probability theory, Topology and Group theory for a beginner.

This year I will start three new 'branch' in mathematics : Probability theory, topology and group theory. I would like to know three complete books i.e. starting with the basics 'tools' whilst going ...
3
votes
1answer
34 views

Constructing a surjection from fundamental group of a mapping cone to Hawaiian Earring to $\prod_\infty \mathbb{Z} / \oplus_\infty \mathbb{Z}$

If X is the subspace of $\mathbb{R}$ consisting of 1, 1/2, ... together with its limit point 0, C is the mapping cone of the quotient map $SX \rightarrow \sum X = $ (the Hawaiian Earring) which ...