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
votes
1answer
16 views

How strong is the operator norm topology?

Let $(V,\tau_V), (W,\tau_W)$ be normable topological vector spaces. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ inducing $\tau_V, \tau_W$ respectively. Let $||\cdot||_{op}$ be the operator norm ...
0
votes
0answers
15 views

Why is $\mathbb{R}^2/G$ homeomorphic to the Klein bottle?

Let $G$ be the group of transformation generated by $a,b:\mathbb{R}^2\to \mathbb{R}^2$ where $a(x,y)=(x+1,y-1)$ and $b(x,y)=(x,y+1)$. We note than $bab=a$ and that $G$ acts properly discontinuously ...
1
vote
1answer
22 views

About the theorem for the first-countable space

I'm learning basic topology in my university. And when learning about the first-countable space, my teacher told us that for every first-countable space, there's a very important theorem like this. ...
0
votes
1answer
29 views

$GL_n(R)$) is open set in $M_n(R)$ [duplicate]

Show that set of all invertible $n\times n$ matrices with real entries (denoted by $GL_n(R)$) is open set in $M_n(R)$. My attempt: by open set I think it means neighborhood of every point in set is ...
-4
votes
1answer
98 views

Topological space that is not sequential and not $T_0$ [on hold]

Construct an example of a topological space $(X,T)$ that is not sequential and is not $T_0$. Preferably the example should not involve a pseudometric, a finite set $X$, or the trivial topology $\{X, ...
4
votes
1answer
73 views

Is $\mathbb R$ connected if we make the rationals open?

Let $\tau$ be the usual topology on $\mathbb R$. Let $\tau'$ be the topology generated by $\tau \cup \{\mathbb Q\}$. Then is the space $(\mathbb R,\tau')$ connected? If so, the it will answer the ...
0
votes
0answers
14 views

Thomsen Blaschke condition

I am reading a paper (Paper 1: https://ideas.repec.org/p/cwl/cwldpp/76.html, that cites another paper ( Paper 2) for its proof. Paper 1, page 1, line 10 says : Consider the topological image G of a ...
1
vote
1answer
13 views

Homeomorphism classes of upper-half spaces without 'boundary' points.

Let $$H=\{(x,y)\in\Bbb R^2: y\geq 0\}$$ i.e. the upper-half space. Let $A$ and $B$ be two finite subsets of the $x$-axis. Then, clearly, $H-A$ is homeomorphic to $H-B$. However, if $A$ and $B$ are two ...
2
votes
2answers
65 views

Baby Rudin Exercise 2.24

I have some difficulties solving the following exercise (Baby Rudin 2.24) Let $ X $ be a metric space in which every infinite subset has a limit point. Prove that $ X $ is separable. In order to ...
5
votes
0answers
25 views

Is a locally countable family of open subsets of an separable space countable?

Definition A topological space $(X, \mathfrak{T} )$ is said to be an separable space if $X$ contains a contable subset $D \subseteq X$ such that $D$ is dense in $X$ Definition Let $(X, \mathfrak{T} ...
1
vote
1answer
36 views

Is the metric ${d(x,y)}\over {1+d(x,y)}$ complete where $d$ is the usual Euclidean metric on $\mathbb R^{2}$

Let $d(x,y)$ be the usual Euclidean metric on $\mathbb R^{2}.$ $\mathbb R^{2}$ is complete under $d(x,y)$. I have this subspace given $$[0,1]\times [0,\infty )\ \ of\ \ \mathbb ...
4
votes
0answers
33 views

Every $f\in\omega^\omega$ is bounded by the “increasing enumeration” of the intersection of a countable dense set and a dense open set in $\mathbb{R}$

I am studying the theorem 2.2.6 of "On the structure of the real line" of book Bartosznky-Judah. In the proof of theorem 2.2.6 the part $(4) \to (5)$ $(4)$ for every family of dense open subsets ...
1
vote
0answers
20 views

Classifying Topological spaces by Kuratowski monoid

I was going through the paper Gardner, Barry J., and Marcel Jackson. ``The Kuratowski closure-complement theorem." New Zealand J. Math, 38 (2008),9--44. It deals with the Kuratowski Closure Complement ...
1
vote
2answers
34 views

Show that $C^{1}([a,b];\mathbb{R})$ is a Banach space.

Let $C^{1}([a,b];\mathbb{R})$ the vectorial space of the functions (bounded) $f:[a,b]\to\mathbb{R}$ where all $f$ has a continuous derivate (and bounded) in all point of $[a,b]$, with the norm ...
3
votes
2answers
58 views

Is the “product topology” a topology?

The question is Let $(\Omega_1,\tau_1)$ and $(\Omega_2,\tau_2)$ be two topological spaces, then is $\left( {{{{\Omega }}_1} \times {{{\Omega }}_2},\tau} \right)$ where $\tau=\{A\times B:A\in ...
0
votes
1answer
25 views

Definition of Roelcke uniformity

Let $(G,\mathcal T)$ be a topological group. The set of all uniformities on $G$ forms a lattice $\frak A$ and the set of all uniformities on $G$ producing $\cal T$, forms a lattice $\frak B$. The ...
1
vote
0answers
18 views

If two embeddings are ambiently isotopic are they isotopic?

Here I am just working in the topological category. I just learned what it means for two embedding $f,g :X \to Y$ to be isotopic - there exists a homotopy $H$ between them such that for all $t$, ...
-1
votes
1answer
52 views

Prove a closed ball is a subset of another ball iff the triangle inequality is true [on hold]

For balls in $\Bbb{R}^n$ prove that: $$\bar{B}(a,r) \subset B(b,s) \iff \|a-b\| \lt s - r.$$
1
vote
0answers
29 views

Maximal compactifications without the Tychonoff theorem

I once saw a neat proof in American Mathematical Monthly of the Tychonoff theorem (The Tychonoff product topology of a family of compact spaces is compact) for the special case of the product of ...
4
votes
1answer
47 views

Does every compact simply-connected subset of $\mathbb{R}^n$ have an efficient $r$-covering path for all $r>0$?

Let $A$ denote a subset of $\mathbb{R}^n$. Definition 0. Given a positive real number $r$, an $r$-covering path of $A$ is a non-negative real number $T$ together with a differentiable function ...
3
votes
1answer
26 views

A strange property of continuous deformations of balls

Let $B$ be the closed unit ball in $\Bbb R^n$ and let $$F:B\times[0,\infty)\to\Bbb R^n,\quad F(x,t)=F_t(x)$$ be a continuous map such that $F_0$ is the identity. In other words, $F$ defines a ...
2
votes
3answers
35 views

Looking for example of topological spaces where sequential continuity does not imply continuity

Please give an example of a function $f : X \to Y $ where $X,Y$ are topological space , such that there exist $x \in X$ such that for every sequence $\{x_n\}$ in $X$ converging to $x$ , $\{f(x_n)\}$ ...
1
vote
1answer
27 views

Understanding this proof about the intersection of compact subsets

The following proof is theorem 2.36 from Rudin's Principles of Mathematical Analysis: Theorem: If $\{K_\alpha\}$ is a collection of compact subsets of a metric space $X$ such that the intersection ...
1
vote
0answers
8 views

Length metric and edge-path metric on a finite dimesional $CAT(0)$ cube complex are coarsely equivalent

I'm trying to find a proof for the statement in the title: Length metric and edge-path metric on the vertex set of a finite dimensional $CAT(0)$ cube complex are coarsely equivalent. Length ...
0
votes
0answers
25 views

Norms on $L(V,W)$

Let $V,W$ be normable topological vector spaces over $\mathbb{F}$. Let $C(V,W)$ be the set of continuous linear transformations $T:V\rightarrow W$. Let $||\cdot||_V, ||\cdot||_W$ be norms on $V,W$ ...
3
votes
1answer
42 views

Closure in a topological product: is AC needed?

I'm working on a proof of $\prod_{\alpha\in\Lambda}\overline{A_\alpha}=\overline{\prod_{\alpha\in\Lambda}A_\alpha}$ in the product topology. This has been asked before, i.e. Closure in a product of ...
6
votes
5answers
76 views

Which of the following condition ensure that the function $f:R^n\to R$ is continuous?

I encountered an interesting problem in my Economics class about continuity. Which of the following conditions on the function $f:\mathbb R^n\to \mathbb R$ ensures that the function $f$ is ...
0
votes
0answers
14 views

equivalent definitions of locally compact space

Let $X$ be Hausdorff space. Equivalent: 1.every point of X has a compact neighbourhood. 2.every point of X has a local base of compact neighbourhoods. the direction $2.\Rightarrow 1.$ is clear. I ...
0
votes
2answers
23 views

Interior points are limit points in $\mathbb{R}$?

I have read another question, and know that interior points are not limit points in general topology space. But when we talk about any subset $\mathbb{A}$ of $\mathbb{R}$, can I say that ...
5
votes
1answer
79 views

Is $SO(n)$ a topological space?

I am reading some articles about covering space in Wikipedia. It says that $\operatorname{Spin}(n)$ is the universal cover of $SO(n)$ for $n>2$. I cannot understand how people view groups as ...
2
votes
3answers
51 views

Boundary of a bounded open set in $\mathbb{R}^2$

Does the boundary of a bounded open set in $\mathbb{R}^2$ necessarily have infinite points? How do we prove that, or is there a counterexample? It seems true to me, but I haven't been able to find a ...
9
votes
0answers
81 views

Is a maximal open simply connected subset $U$ of a manifold $M$, necessarily dense?

There is a short argument using Zorn's lemma and the compactness of $[0,1]$, that shows every manifold must have maximal open simply connected subspaces. However, I am wondering if it is necessarily ...
-2
votes
0answers
25 views

Find a metric space $(X,d)$, such that $\partial B_r(x)\neq S_r(x)$, where $S_r(x)$ ={ y $\in$X: $d(x,y)=r$} and $B_r(x)$ ={ y $\in$X: $d(x,y)<r$} [on hold]

Find a metric space $(X,d)$, such that $\partial B_r(x)\neq S_r(x)$, where $S_r(x)$ ={ y $\in$X: $d(x,y)=r$} and $B_r(x)$ ={ y $\in$X: $d(x,y)<r$}
1
vote
2answers
30 views

Question about product topology notation

Instead of using the general form, I will use a simpler one such as $\mathbb{R} \times \mathbb{R}$ (which is $\mathbb{R}^2$ of course). Now the notation says that the open sets are the union of the ...
0
votes
2answers
23 views

Show that the projection map $p: \mathbb{R}^2 \to \mathbb{R}$, where $p(x,y) = x$, is open

I have to take an open set in $\mathbb{R}^2$ and show that it maps to an open set in $\mathbb{R}$. So let $A \times B$ be an open set in $\mathbb{R}^2$. I have to show that $A$ is an open set. By ...
-1
votes
0answers
26 views

On classification of directed topological spaces [on hold]

Is classification of directed topological spaces (not their homotopy equivalence classes!) an important subject in modern mathematics?
7
votes
4answers
110 views

The Galois connection between topological closure and topological interior

[Update: I changed the question so that $-$ is only applied to closed sets and $\circ$ is only applied to open sets.] Let $X$ be a topological space with open sets $\mathcal{O}\subseteq 2^X$ and ...
2
votes
1answer
45 views

Understanding Rudin's proof that compact subsets of metric spaces are closed.

Rudin's Principles of Mathematical Analysis has the following definition of compact: A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. ...
0
votes
0answers
34 views

Can a linear projection of spheres be a torus?

Assume that we have two disjoint subsets $A_1, A_2 \in \mathbb{RP}^4$ that are both homeomorphic to the sphere $S^2$. Let $\pi$ be the linear projection with centre a point that does not lie on $A_1$ ...
1
vote
1answer
26 views

Finding an element in $l_1$ space with certain properties

I am facing a bit problem in the following: Given $x_1,...,x_m \in l^\infty$ and positive $\epsilon_1,...,\epsilon_m$, I need to find an element $a= (a_n)$ in $l_1$ space such that $\sum_{n=1}^ \infty ...
1
vote
2answers
43 views

$f: \mathbb{R} \to S^1$ where $f(x) = (\cos x, \sin x)$ open and closed mapping?

Show that $f: \mathbb{R} \to S^1$ where $f(x) = (\cos x, \sin x)$ is both an open and closed mapping, or provide counter-examples if one or both are not true. Well, my hypothesis is that they are ...
0
votes
0answers
53 views

rudin's definition of a compact set

Here are some definitions given in my book: Definition 2.31 By an open cover of a set $E$ in a metric space $X$ we mean a collection $\{G_\alpha \}$ of open subsets of $X$ such that $E \subset ...
0
votes
2answers
25 views

Can we classify all spaces which go by the given below problem

In chat I was discussing this problem which I thought of while doing my revision: If $M$ is a subspace of the space $X$ and we have a mapping of $M$ from the space $Y$ can I extend this map to a ...
1
vote
1answer
33 views

Any homeomorphism from $D^2$ to $D^2$ maps $\partial D^2$ onto $\partial D^2$

I'm starting to study Algebraic Topology. After doing some problems and studying the theory I've arrived at: Let $D^2$ be the unit disk in $R^2$, $\partial D^2$ the topological boundary of $D^2$ ...
9
votes
7answers
502 views

What does it REALLY mean for a metric space to be compact? [duplicate]

I've been trying to wrap my head around the concept of compactness and get an intuitive understand of what it is. The definition used in my text book is the finite subcover definition. A subset ...
0
votes
1answer
31 views

Question about “subset of topological space”

In the Topology book I'm studying, there is a exercise that starts off with the statement: Suppose $X$ is a topological space, $A$ is a subset of $X$... I'm not 100% sure what this means. First ...
1
vote
1answer
34 views

Show the “clock”and Euclidean metrics generate different topologies

I'm trying to teach my self topology. I wanted to find an example of a metric generating different topology. I came up with what a call "clock" metric, inspired by the modulo operation. Can anyone ...
2
votes
0answers
26 views

Int M is open and a manifold

If M is an n-dimensional manifold with boundary, then Int M is an open subset of M , which is itself an n-dimensional manifold without boundary. I am supposed to use these definitions: If M is an ...
-1
votes
1answer
48 views

Give the example of compact set with infinite countable derived set [on hold]

Can anyone give me an example of compact set of which the derived set is infinitely countable set?? thks in advance, I have no idea about this .
4
votes
1answer
60 views

Is the set open?

Define a complex polynomial $p:\mathbb{C}\longrightarrow\mathbb{C}$ where $\deg p=n\in\mathbb{N}$. \begin{equation} p(z) = \alpha_{n}z^{n}+\alpha_{n-1}z^{n-1}+\dots+\alpha_{1}z+\alpha_{0},\quad ...