Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions tagged [general-topology]

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; ...

0
votes
0answers
7 views

proving $(\mathbb{B}^2 \times \{0 \}) \bigcup (\mathbb{S}^1 \times [0, \infty))$ is a retract of $\mathbb{R}^3$

I don't know how to solve this. I tried constructing a retraction but nothing comes to mind. Can someone guide me through this and if possible explain the intuition behind solving this kind of problem?...
0
votes
3answers
23 views

Difference between metric spaces and metrizable spaces (Munkres)

Is there a difference between a metrizable space and a metric space ? For me if $X$ is a metrizable space, then there is a metric $d$ s.t. $(X,d)$ is a metric space. And obviously a metric space is ...
1
vote
2answers
25 views

If $X$ and $Y$ are subspaces of $Z$, $X \cong Y$ and $X$ is a retract of $Z$, is $Y$ also a retract of $Z$?

If $X$ and $Y$ are subspaces of $Z$, $X \cong Y$ and $X$ is a retract of $Z$, is $Y$ also a retract of $Z$? I think the answer is no, but I can't find a counterexample. Can anyone help me with this?
10
votes
5answers
354 views

Interesting questions(with answers) about concepts in topology for an amateur audience

I have been asked to hold an introductory math quiz for the Freshmen batch in my college. It entails interesting questions about different areas of mathematics presented in such a way so that it seems ...
1
vote
1answer
19 views

Nontrivial example of closed set relative to subspace but not with respect to original space

I am interested in finding the example of the set which is close relative subspace but not in original space. I know that $\mathbb Q $ is the subspace of $\mathbb R$ And $\mathbb Q$ is close relative ...
2
votes
3answers
33 views

Bases and pseudo-bases in topological spaces

Let $X$ be a topological space and $x \in X$. A base for $X$ at $x$ is a collection $\mathcal{B}$ of open sets of $X$, all of them having $x$ as one of its elements such that for every open ...
0
votes
0answers
23 views

Relationship between universal vector bundles and principal $O(n)$ bundles?

I am wondering if knowing a model for the universal principal $O(n)$ bundle would allow one to infer a corresponding model for the universal rank $n$ vector bundle, say via a balanced product ...
1
vote
0answers
17 views

Mapping Class Group of Knot Complements and Their Heegaard Splitting

Using this algorithm and the ideas from this paper (page 3) I gather that I can present $S^3-K$ where $K$ is the figure eight knot as the gluing of two genus 5 handle bodies, $H_5$, i.e. $$ S^3-K=H_5\...
3
votes
0answers
28 views

Does strong convergence in probability of conditional probabilities imply countable additivity on open sets?

Please note. This question is a continuation of another question of mine. I'd really prefer hints to complete answers for this one. Let $\Omega = \{0,1\}^\mathbb{N}$, and let $\mathcal{B}$ be the ...
1
vote
1answer
25 views

$\pi_3(Sp(N))=\mathbb{Z}$, $\pi_4(Sp(N))=\mathbb{Z}_2$, $\pi_5(Sp(N))=\mathbb{Z}_2$?

From the computation of some lower dimension $N$ of $Sp(N)$ group, we see that the homotopy groups are: $\pi_3(Sp(N))=\mathbb{Z}$, $\pi_4(Sp(N))=\mathbb{Z}_2$, $\pi_5(Sp(N))=\mathbb{Z}_2$, at ...
0
votes
1answer
41 views

Countably Additive Set Function Dealing with Extended Real Numbers

Define $n(E):=\left\{\begin{matrix} \infty,& \text{if E is a not finite set} \\ |E|,& \text{if E is a finite set.} \end{matrix}\right.$ Let $\lbrace E_i\rbrace$ be an arbitrary sequence ...
0
votes
1answer
32 views

Cluster points and equality of sets

I am trying to understand a proof of the following result: If $A $ is a subset of a fixed set $V $ in $\mathbb R^n $ then the boundary of $A $ in $V$ is $V\cap \partial A $. The proof runs as ...
0
votes
3answers
36 views

If $\varphi:\Omega \to \mathbb R^n$ and $\Omega\subset \mathbb R^n $ not supposed open, does $\varphi : \Omega \to \varphi(\Omega )$ a homeomorphism? [on hold]

Let $\Omega \subset \mathbb R^n$ and $\varphi:\Omega \to \mathbb R^n$ an injective and continuous function. I know that if $\Omega $ is open then $\varphi(\Omega )$ is open an $\varphi:\Omega \to \...
2
votes
1answer
26 views

If $\varphi:V\to \mathbb R^n$ continuous and injective, why $\varphi|_{\bar B}: \bar B\to \varphi(\bar B)$ an homeomorphism ?

Let $\Omega \subset \mathbb R^n$ open and $$\varphi:\Omega \to \mathbb R^m$$ continuous and injective. We suppose $m>n$. If $B\subset \bar B\subset \Omega $ is a ball, why $$\varphi|_{\bar B}: \bar ...
0
votes
0answers
31 views

Proof of partition of unity in Spivak's Calculus on manifolds

In Case 3 of Spivak's proof of partition of unity, he seems to assume the existence of boundary of $A$. I wonder if there is a case where the boundary $A$ is $\Phi$, which requires an extra proof? ...
-1
votes
1answer
29 views

Minimally dense subsets of $[0,1]$? [duplicate]

I'm curious about the notion of a minimally dense subset of a given space (I've been using $[0,1]$, but if others are interesting, I'm interested in that too!). Here are two questions. Does there ...
0
votes
1answer
28 views

continuous function and uniform convergence

Let $X$ be a topological space and $Y$ be a metric space. Let $f_n:X\rightarrow Y$ be a sucession of continued functions on $X$. If $(f_n)$ converges uniformly to $f \Rightarrow f$ is continuous. ...
1
vote
3answers
29 views

Prove: $F=\{x_n\}\cup \{x\}$ Is closed [duplicate]

Let $M$ be a metric space, $x_n\in M$ a sequence which converges to $x\in M$ Prove: $F=\{x_n\}\cup \{x\}$ is a closed set So we have $x_n\to x$ such that $x_n\in F$ and $x\in F$ and we know that a ...
0
votes
3answers
43 views

Can any metric on $\Bbb R^n$ be bounded above and below for any other metric? [duplicate]

Let $d_1(x,y)$ and $d_2(x,y)$ be any two metrics on $\mathbb{R}^n$. Can it be shown that, $$c\cdot d_2(x,y) \le d_1(x,y) \le C\cdot d_2(x,y)$$ for all $x,y \in \mathbb{R}^n$ for some fixed positive ...
1
vote
1answer
33 views

A closed set $A\subset Y$ is closed on $Y$ iff $\pi^{-1}(A)$ is closed on $X$.

Let $X$ be a topological space, $Y$ be a quotient space of X and $\pi:X\rightarrow Y$ be a quotient function. How to prove the following: A closed set $A\subset Y$ is closed on $Y$ iff $\pi^{-1}(A)$ ...
0
votes
0answers
15 views

Translation in one-point compactification

I'm trying to determine when translating a subset of a space is homeomorphic to the original space. This obviously isn't always possible since connectedness is a topological property. The motivation ...
0
votes
1answer
24 views

A local homeomorphism is open.

Let $\Omega \subset \mathbb R^n$ an open and $\varphi:\Omega \to \mathbb R^n$ a local homeomorphism. Prove that $\varphi$ is an open application, that mean that $\varphi(U)$ is open for all $U\subset \...
0
votes
1answer
24 views

Continuous function maps connected set onto connected set

Let $X, Y$ be topological spaces and $f: X\to Y$ continuous. Show that $f$ maps connected sets onto connected sets. I am not sure if I understand the definition of a connected set right. Our ...
10
votes
4answers
738 views

Topologically, is there a definition of differentiability that is dependent on the underlying topology, similar to continuity?

I'm studying Analysis on Manifolds by Munkres, and at page 199, it is given that Let $S$ be a subset of $\mathbb{R}^k$; let $f: S \to \mathbb{R}^n$. We say that $f$ is of class $C^r$, on $S$ if $...
0
votes
1answer
37 views

Isn't there any meaning to talk of open spaces, or of closed space?

From Rudin's Principles of Mathematical Analysis (p.37). Theorem 2.33. Suppose $K\subset Y\subset X$. Then $K$ is compact relative to $X$ if and only if $K$ is compact relative to $Y$. Immediately ...
2
votes
1answer
41 views

Application of connectedness [on hold]

I am reading connectedness in topology. Wondering where we can use this ? I have searched on internet but found only its role in motion planning robotics, population modeling.
0
votes
0answers
12 views

Prove that $\exists \;!\;g:E_1\to E_2,$ uniformly continuous such that $g$ is an extension of $f.$ [duplicate]

Let $(E_1,d_1)$ and $(E_2,d_2)$ be metric spaces. Let $D\subset E_1$ be dense in $E$. Let $f:D\to E_2$ be uniformly continuous. If $(E_2,d_2)$, then $\exists \;!\;g:E_1\to E_2,$ continuous such that $...
2
votes
2answers
46 views

Prove that there exists a continuous function $f:E\to \Bbb{R}$ such that $f(x)=1$ if $x\in A$ and $f(x)=0$ if $x\in B.$

Let $A$ and $B$ be closed sets in a metric space $(E,d)$ such that $A\cap B=\varnothing$ Prove that there exists a continuous function $f:E\to \Bbb{R}$ such that $f(x)=1$ if $x\in A$ and $f(x)=0$ if $...
4
votes
1answer
26 views

Discrete subset of a separable and normal space is countable

Let $X$ be separable and normal. Suppose $A\subset X$ is discrete in the relative topology. I'd like to show that $A$ is countable. By separability, it suffices to exhibit a mutually disjoint ...
2
votes
0answers
40 views

What are these Moebius-esque strips called?

Suppose we have a ring-like shape with two broad faces... We cut the ring and make a twist of 180 degrees to produce the world famous Moebius strip. A one-faced shape. Let's say we cut the shape ...
0
votes
1answer
59 views

Prove that $\{x\in \Bbb{R^n} :\;f(x)\leq k\}$ is compact

Let $f:\Bbb{R^n}\to \Bbb{R}$ be continuous such that $$\lim_\limits{||x||\to\infty}f(x)=+\infty.$$ I want to show that $\{x\in \Bbb{R^n}:\; f(x)\leq k\}$ is compact. I'm thinking that it suffices ...
3
votes
1answer
39 views

Every point of the unit circle is a limit point of $x_k = \left(1+\frac{1}{2^k}\right)\begin{bmatrix} \cos k\\\sin k\\ \end{bmatrix}$

Consider the function $f:\mathbb{R}^n\to \mathbb{R}$ defined by $f(x) = |x|^2$ Show that $f(x_{k+1})<f(x_{k})$ for $$x_k = \left(1+\frac{1}{2^k}\right)\begin{bmatrix} \cos k \\ ...
1
vote
2answers
38 views

Why $\mathbb R$ with the metric $d(x,y)=\min\{|x-y|,1\}$ is not compact?

Refer to Heine-Borel theorem, since $\mathbb R$ is a vector space of finite dimension, all closed and bounded of $\mathbb R$ should be compact. Here $\mathbb R$ is closed and bounded, therefore it ...
0
votes
3answers
48 views

Difference between closed under countable union and arbitrary union

From my research, I am confused in differentiating the difference between closed under arbitrary unions and countable unions. What exactly is the difference between them? In an example, it states ...
1
vote
2answers
39 views

every open set in R is the union of an at most countable collection of disjoint segments

I have some doubts in proof. R is separable, and thus has a countable dense set, namely Q. Let G ⊂ R be any open set. Then Q∩G is a countable dense set in G by the Archimedean property, and since G ...
0
votes
0answers
41 views

Continuous bijection is homeomorphism

(I was able to fix my question while I was writing) I am asked to show: Let $X,Y$ be topological spaces with $X$ compact and $Y$ Hausdorff. Let $f:X\to Y$ be continuous and a bijection. Then is $f^...
0
votes
1answer
36 views

Prove path independence of $\int_{\gamma} f$ w/ weaker conditions on $f$

A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Ch5.2 2 Questions about Cor 5.8 and Cor 5.9 (*) Question 1. Can we prove Cor 5.9 using Cor 5....
0
votes
1answer
22 views

Show that a proper locally invertible map is surjective

Let $f:R^2\to R^2$ be a continuously differentiable function such that $Df(x)$ is invertible for all $x\in R^2$ and $f^{-1}(K)$ is compact for every compact set $K$. Show that $f$ is surjective. The ...
1
vote
1answer
67 views

Example of a CW complex that is not a $\Delta$-complex?

Hatcher notes in chapter 2.1 (in the paragraph just preceding the section on simplicial homology (page 104 in my edition)), that all $\Delta$-complexes can be realized as CW complexes with some added ...
3
votes
2answers
65 views

Given $k$ points in $n$-dimensional space, is there always a continuous $n-1$ surface that can divide the points into two arbitrary groups?

Say we have $k$ points in set $P\mid x_i\in\mathbb{Z}^n$, such that $k=\mathcal{O}(n!)$. We now arbitrarily divide the points into two sets, $A$, $B$. Note that $A\cup B = P$ and $A \cap B = \...
0
votes
3answers
50 views

Topology definition: finite intersections vs. infinite unions

In the definition of topology, we allow infinite unions but only allow finite intersections. As mentioned by many other answers (see In a topological space, why the intersection only has to be finite?;...
2
votes
0answers
51 views

$\mathbb C^{\infty} / \{0\}$ retracts on $S^{\infty}$

We want to show that $\mathbb C^{\infty} / \{0\}$ retracts on $S^{\infty}$. $C^{\infty} / \{0\}$ should be the same as $\mathbb R^{\infty} / \{0\}$. Maybe the retraction could be $(1-t)x + t \frac{x}{...
3
votes
2answers
34 views

Does any isomorphism between $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$ always induce a homeomorphism between $(X,x_0)$ and $(Y,y_0)$?

I know that if $h : (X,x_0) \longrightarrow (Y,y_0)$ is a homeomorphism then that induces an isomorphism $h_{*} : \pi_1(X,x_0) \longrightarrow \pi_1(Y,y_0)$ defined by $$h_{*} ([f]) = [h \circ f].$$ ...
0
votes
2answers
39 views

Compact set in normed vector space of infinite dimension

I know that in metric spaces, compact set are set such that every sequence has a convergent subsequence. As a surprising fact of a question I asked here, in infinite dimension, balls are not compact. ...
1
vote
1answer
31 views

Proper function equal to $1$ in a closed subset

Let $\Omega \subset \mathbb R^n$ be an open subset and $F\subset \Omega$ a (relatively) closed set. Suppose that the projections $\pi_x, \pi_y:\Omega\times\Omega \longrightarrow \Omega$ are proper in $...
2
votes
1answer
24 views

Alternative characterizations of the strong topology in a normed linear space

Let $X$ be a normed linear space. The strong topology on $X$ is the topology induced by the norm, i.e. it has the open balls $B_r(x)$ as its basis. Is there an alternative characterization of ...
5
votes
1answer
56 views

Is the given set open ball in metric space or not?

Metric space is $(C[0,1],||.||_\infty)$ with sup norm Let $f,g:[0,1]\to R$ be continuous functions and $f(t)<g(t)\forall t\in [0,1]$. $A=\{h\in C[0,1]|f(t)<h(t)<g(t) \forall t\in [0,1]\}$. ...
0
votes
1answer
13 views

Question on invariance of domain of Brouwer

Let $\Omega \subset \mathbb R^N$ and $\varphi:\Omega \to \mathbb R^n$ an injective and continuous function. Brouwer proved that $\varphi$ is an open application and that $\varphi$ is a homeomorphism. ...
0
votes
1answer
28 views

Let U be open in $\Bbb{R}^n$ such that $U\neq \emptyset$ . Prove that if U is complete, then $U=\Bbb{R}^n$

I have this interesting problem. Let U be open in $\Bbb{R}^n$ such that $U\neq \emptyset$ . Prove that if U is complete, then $U=\Bbb{R}^n$. According to my lecturer, this could be a one-line ...
5
votes
1answer
26 views

Converse part of triangular inequality hold equality when third point lies in the both point

I was reading following proof. In that I understand first part But In converse part I do not understand Why if equality hold then d(x,z) is scalar times d(z,y)? As shown in yellow highlight. I do not ...