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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3
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1answer
89 views

Prob 9, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to prove the generalised tube lemma?

The tube lemma is as follows: Let $X$ and $Y$ be topological spaces. Let $Y$ be compact. Let $x \in X$. If $N$ is an open set in $X \times Y$ such that $x \times Y \subset N$, then there is an open ...
1
vote
1answer
36 views

Diffeomorphism between vector bundles

I have some difficulty solving the following problem: Let $M$ be a diffentiable manifold of dimension $m$, which admits a global base of differentiable vector fields $\{X_1,\ldots,X_m\}$; this ...
1
vote
4answers
100 views

Sum of open/closed/compact sets in $\mathbb{R}^n$ open/closed/compact

I know that the following exercise you can find on internet maybe with solution too, but I want to know, if my "solutions" are correct. Let $X,Y\subset \mathbb{R}^n$, $X+Y=\{x+y;x\in X, y\in Y\}$. ...
3
votes
2answers
135 views

Complementary compactness

Let $X$ be a topological space having the property that whenever a subset $A$ of $X$ is compact, then $X\setminus A$ is compact too. Is every subset of $X$ compact?
0
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1answer
36 views

Relation between open, semi-open and preopen

Does any one of the semi,pre and just open given by definitions below imply another one under no assumption about the space? If $(X, \tau)$ is a topological sapce and $A \subset X$, then $A$ is ...
4
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3answers
53 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of X. If $A\subseteq B$ then $A' \subseteq B'$

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of X. If $A\subseteq B$ then $Bd(A) \subseteq Bd(B)$ If $A\subseteq B$ then $A' \subseteq B'$ ($A'$ is the set ...
1
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0answers
89 views

The skeleton of Eulerian polyhedra

There is (at least) two kind of validity domain of Euler's $v−e+f=2$ polyhedron formula. One is the "Eulerian" polyhedra, i.e. simply connected polyhedra with simply connected faces (see here). The ...
2
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1answer
50 views

What does it mean: $a\in X$ is open in $X \subset \mathbb{R^n}$

I'm in a course of multivariable real analisys and I have to prove this: $a\in X$ is open in $X\subset\mathbb{R}^n$ (in the related topology to $X$) if and only if $a$ is a isolated point. I ...
4
votes
2answers
65 views

Let $A$ be a closed subset of a connected top space $X$. If the $\text{Bd }A$ is connected, does it imply $A$ is also connected?

I came across a question from Munkres: Problem 24.11: If A is a connected subspace of X, does it follow that IntA and BdA are connected? Does the converse hold? Thinking about the converse, I know ...
5
votes
1answer
91 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why ...
2
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1answer
59 views

Characterization of closed map by sequences/nets

I'm interested in characterizing closed maps in terms of nets. Since a map is closed iff $\overline{f(V)} \subseteq f(\overline{V})$ for all subsets $V$, I believe one possible such characterization ...
5
votes
1answer
126 views

What is the topology of the hyperreal line?

Denote by $\Bbb R$ the real line and by $\Bbb R^*$ the hyperreal line. For any real numbers $x < y < z$ and infinitesimal $\epsilon$ the following holds: \begin{equation} \forall a,b,c \in \Bbb ...
0
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0answers
25 views

Necessity of continuity in Topological Vector Space

In the notion of a topological vector space, we define such as a vector space $X$ (over a field $\mathbb{K}$) with topology $\mathscr{T}$ such that $$\iota_+: (X,\mathscr{T}) \times (X, \mathscr{T}) ...
4
votes
1answer
24 views

Is this proof involving complete metric spaces correct?

Show that if every closed ball of a metric space $(X, d)$ is complete then $ X$ is complete. I thought the following: given $(x_n)$ a Cauchy sequence in $X$, we have that the set $A= \{x_{1}, ...
0
votes
1answer
29 views

The definition of C^r Structural Stability

I currently have a definition that states that given a flow $f$, $f$ is structurally stable if for any $g$ in some neighborhood of $f$, $f$ and $g$ are topologically conjugate. Would the definition ...
4
votes
0answers
28 views

What does it mean for a function f to be C^k “close” to a function g?

My impression is that $f$ is $C^k$ "close" to $g$ if $f$ is close to $g$, $f'$ is close to $g'$, ... $f^{(k)}$ is close to $g^{(k)}$. However, my professor is being a bit nebulous on what "close" ...
5
votes
1answer
92 views

Is the set of probability measures on a compact metric space (weak*-)closed?

Let $(S,\mathcal B)$ be a compact metric space with the Borel-$\sigma$-Algebra. Let $\mathcal M$ be the space of signed Borel measures and $\mathcal P \subset \mathcal M$ the set of probability ...
2
votes
1answer
62 views

Showing this function is continuous $ f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I ...
4
votes
1answer
33 views

Given a fixed path connected topological space $X$, is “$Y$ is homotopy equivalent with $X$” always strictly weaker than “$Y \approx X$”?

$\simeq$ will denote homotopy equivalence, and $\approx$ will denote homeomorphism. Given many spaces, it is easy to show this property, for example using the fact that $X \times I \simeq X$, where ...
1
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1answer
37 views

$[0,1]\times\mathbb{N}/(0,k)$ not metrizable.

$X = [0,1]\times\mathbb{N}/((0,1)\sim(0,2)\sim\dots)$. I read that $X$ is not metrizable since sequence $\{(\frac{1}{n},n)\}$ is closed in $X$ and therefore does'n have limit. But i don't understand ...
2
votes
2answers
156 views

Give examples of compact spaces $A,B$ such that $A\cap B$ is not compact

If a topological space is Hausdorff then arbitrary intersection of compact sets is compact. How to find examples of compact subsets $A,B$ of a topological space $X$ such that $A\cap B$ is not ...
4
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1answer
42 views

Coincidence points on compact Hausdorff spaces.

I am really stuck on this exercise in my course notes. Let $X$ and $Y$ be compact Hausdorff spaces and $f, g : X \to Y$ be continuous functions. Show that: There is an $x \in X$ with $f(x) = ...
1
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1answer
27 views

Two questions about increasing unions of compact subsets of a locally compact Hausdorff group.

I have two questions to ask related to my research. Question 1. Let $ G $ be a locally compact Hausdorff group. Is it possible that $ G $ is the union of a chain of compact subsets (ordered by ...
0
votes
1answer
58 views

multiplying two metrics

Let $(X,d)$ and $(X,d^\prime)$ are metric spaces ,is $d×d^\prime$ metric on $X$ ?I try to prove triangle inequality , I write two triangle inequalities for $d $ and $ d^\prime$ but it not true.
0
votes
1answer
31 views

injective/path component

For $f$ to be injective it'll have to be one to one, so could i somehow create a field $X$ which is one to one, however have it so somehow the path component for $X$ is $2$ and the path components ...
0
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1answer
67 views

Examples of homotopy [closed]

I Understand the definition of homopoty and understand somewhat how to apply it. However, i find it difficult to give examples
5
votes
1answer
72 views

Cardinality of order topology?

Just out of interest: The cardinality of the Euclidean topology on the real line is $c$. In general, if $X$ is totally ordered of cardinality $\alpha$, the order topology on $X$ must have cardinality ...
0
votes
0answers
48 views

How to prove that the topological spaces are homeomorphic

Let's consider a topological space $X$. We define $X \times I / {\sim}$ as a product of $X$ and a $I=[0, 1]$ closed interval's quotient by the following equivalence relation: $(x, 1) \sim (y, 1), (x, ...
1
vote
3answers
70 views

Intersection of open sets in $\mathbb{R}$

Give examples of the following with justification. A collection of open subsets $(U_n)_{n\in\Bbb N}$ of $\Bbb R$ such that $\bigcap\limits_{n\in\Bbb N}U_n$ is not open. as far as I'm aware the ...
3
votes
1answer
137 views

Is this a compact space?

Let $A=\{x:d_\infty(x,0)\le 1 \}$, the subspace of the space of bounded sequences $x=(x_n)^\infty_{n=1}$, $x_n\in \mathbb{R}$, with metric $\{x:d_\infty(x,y)= sup_n |x_n-y_n| \}$. The answer says it ...
2
votes
3answers
41 views

Conceptual problem regarding distance between two sets.

Given a metric space $(X,d)$ and two non empty subsets $A,B \subset X$ we define the distance between $A$ and $B$ as $$ d(A,B) = \inf\, \{d(a,b) : a\in A, b\in B\} $$ My question is the following: ...
1
vote
2answers
44 views

interior points having some distance from the boundary.

Let $U \subset \Bbb R^n$ be open, and let $U_\epsilon:=\{ x\in U | d(x,\partial U)>\epsilon\}$. Is it true that $(U_\epsilon)_\delta=U_{\epsilon+\delta}$? I tried to prove it, but I couldn't, and ...
1
vote
1answer
25 views

Path components

I have drawn the two different functions and I can fully see that $|\pi_0(X)|=2$, But how would I show this? I'm assuming I have to show something on the lines that $U_+$ and $U_-$ alone are both ...
1
vote
1answer
58 views

Example 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How to show this set to be open?

Let $N$ be the following subset of $\mathbb{R}^2$: $$N \colon= \{ \ (x,y) \in \mathbb{R}^2 \ \colon \ \vert x \vert < \frac{1}{y^2+1} \ \}.$$ Then intuitively it is apparent that $N$ is open. ...
1
vote
1answer
28 views

Prob. 10 (b), Sec. 25 in Munkres' TOPOLOGY, 2nd ed: How to show that components and quasicomponents are the same for locally connected spaces?

Let $X$ be a topological space; let us define $x \sim y$ if there is no separation $X = A \cup B$ of $X$ into disjoint open sets such that $x \in A$ and $y \in B$. This relation is an equivalence ...
2
votes
2answers
64 views

Find all covering spaces of $\mathbb{RP}^n \times \mathbb{RP}^n$, $n>1$

Let $X = \mathbb{RP}^n \times \mathbb{RP}^n$. I know the following: the universal cover of $X$ is $Y = \Bbb S^n \times \Bbb S^n$ the fundamental group of $X$ is $G = \Bbb Z/2 \Bbb Z \times \Bbb Z/2 ...
2
votes
1answer
54 views

Lusin space, isolated point

I have a question about Lusin space. Definition A Hausdorff topological space, $(X,\tau)$ is said to be a Lusin space if, there exists a topology $\tau'$ (on $X$) stronger than $\tau$ such that ...
2
votes
1answer
25 views

Equivalence of sequence spaces

Let $m$ be the space of infinite sequences $(x_i), |x_i| \leq 1$ with norm $\sup_{i>0}|x_i|$. Let $\ell$ be the space of infinite sequences $(x_i), \sum_{i> 0}|x_i| \leq 1$ with norm $\sum_{i ...
1
vote
0answers
32 views

Prob. 5, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Is there a connected set that is locally connected at none of its points?

Let $A$ denote the rational points of the interval $[0,1] \times 0$ of $\mathbb{R}^2$. Let $T$ denote the union of all line segments joining the point $p = 0 \times 1$ to points of $A$. Then I can ...
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vote
1answer
47 views

Prove that the complement of an open ball in $\mathbb{R^n}$ has exactly one unbounded component [duplicate]

Question: Let $B^n \subset \mathbb{R^n}$ be open ball in the Euclidean metric. Prove that the complement of $B^n$ in $\mathbb{R^n}$ has exactly one unbounded component (components of a set are class ...
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votes
1answer
53 views

Prob. 3, Sec. 25 in Munkres' TOPOLOGY, 2nd ed: Is $I \times I$ path connected or locally path connected in the subspace topology?

Let $$I \ = \ [0,1] \ = \ \{\ x \ \in \mathbb{R} \ \colon \ 0 \leq x \leq 1 \ \}. $$ In the subspace topology that $I \times I$ inherits from the dictionary order topology on $\mathbb{R} \times ...
0
votes
1answer
37 views

Is a linear span of finite set from a finite dimensional space topologically closed?

Let $S=\{x_1,\ldots,x_m\} \subset \mathbb{C}^n $ is it true that: $$ Span (S) = \overline{Span (S)} $$ Must we assume both of the following assumptions? or one of them will be enough? The spanning ...
1
vote
1answer
75 views

is the homomorphism induced by the inclusion map is the inclusion map?

let $X$ be a topological space, $A\subset X$ subspace. Consider $i:\:A\to X$ the inclusion, and $i_* :\:H_n(A)\to H_n(X)$ the induced homomorphism. is $i_*$ the natural homomorphism $[a]\mapsto ...
2
votes
1answer
23 views

How do i prove that $T(A_1)\geq T(A_2)$ iff $A_1$ is a subset of $A_2$?

Let $X$ be a non-void set and let $A$ be a subset of $X$. Denote $T(A)$ the topology on $X$ whose open sets are the null-set and the sets $O$ which contain $A$. show that We have $T(A_1)\geq T(A_2)$ ...
1
vote
1answer
60 views

Proving that a punctured disk is not simply connected, using a specific definition

I am dealing with the same set based on my previous question. I want to show that the set $H = \{z \in \mathbb{C} : 0 < |z| < 1\}$ is NOT simply connected, using the following definitions, that ...
0
votes
1answer
58 views

Why is the inclusion an isomorphism?

Consider $X$ a path-connected space, $A\subset X$ a non-empty subset. My textbook makes the following claim without any explanation, and I wondered if you could help: it says that the inclusion $H_0 ...
0
votes
1answer
48 views

Prove that if $S\subset \mathbb{R}^n$ is not countable, then there exists $x \in S$ such that $x$ is a condensation point.

Let $S \subset \mathbb{R}^n$ with the usual metric. A point $x \in \mathbb{R}^n$ is said to be a condensation point of $S$ if for all $r>0$, $B(x,r)\cap S$ is not countable. Show that if $S$ is ...
3
votes
2answers
109 views

intuition on the fundamental group of $S^1$

I am familiar with the proof that the fundamental group of the unit circle $S^1$ is $\mathbb Z$, yet I couldn't develop intuition for why it is true. For example, why would I fail if I try to find ...
1
vote
1answer
48 views

Is every homeomorphism of $\mathbb{Q}$ monotone?

It is well known that every continuous injective map $\mathbb{R}\rightarrow\mathbb{R}$ is monotone. This statement is false for maps $\mathbb{Q}\rightarrow\mathbb{Q}$. (That is becaus $\mathbb{Q}$ is ...
0
votes
1answer
45 views

Prove $Y$ is closed in $X$ given that $(X, d)$ is a metric space and $Y$ is a subspace of $X$ such that the induced metric is the discrete metric.

So far I have: Since no point $P$ is a limit point of any set except $\{P\}$ with the discrete metric, therefore any given set that is part of $Y$ contains all its limit points, and hence $Y$ itself ...