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
2answers
20 views

Is it true that finite intersection distributes over arbitrary unions?

I have come across the problem of showing that $$\bigcap_{i=1}^n \Big ( \bigcup_{\alpha\in A} X_\alpha^{(i)}\Big) = \bigcup_{\alpha\in A} \Big ( \bigcap_{i=1}^n X_\alpha^{(i)} \Big)$$ for some family ...
0
votes
1answer
13 views

Gluing topological maps and considering the preimage

Given topological spaces $X$ and $Y$ and an open cover $\{W_i\}_{i\in I}$ of $X$. Suppose for each $i$ there exists a continuous map $f_i\colon W_i\to Y$ such that the restrictions of $f_i$ and $f_j$ ...
2
votes
5answers
50 views

Continuous maps in topology; the definition?

I am just wondering, given the definition of continuous maps as follows, A functionn $f:X \to Y$ is continuous if for every open subset $U $ of $Y$ the preimage $f^{-1}U$ is open in $X$. I guess ...
0
votes
1answer
41 views

is this map an homotopic equivalence of pairs from (disk, sphere) to the disk seen attached to a space?

Hello I was studying https://www2.warwick.ac.uk/fac/sci/maths/people/staff/vincent/cohomology.pdf On page 18 given a space $A$ and a map $f:\mathbb{S}^{n-1} \rightarrow A$ he defines the cone $X := ...
1
vote
0answers
14 views

Is $\mathbb{R}^n$ connected under the density topology?

Let $\mu$ denote Lebesgue measure and define the function $ d : \mathbb{R}^n \times \text{PS}(\mathbb{R}^n) \to [0,1] $ by $$ d(x,S) = \lim_{r \to 0^+}{\frac{\mu(B(x,r)\cap S)}{\mu(B(x,r))}}. $$ We ...
4
votes
4answers
178 views

Provide examples or explain why it is impossible

a) A continuous function defined on an open interval with range equal to a closed interval. My example: $f(x)=\frac{1}{2}\sin(4\pi x)+\frac{1}{2}$ on $(0,1)$ to $[0,1]$. Note: I am not considering ...
1
vote
1answer
25 views

Strong equivalence between Lévy’s metric and a topologically equivalent metric

Let $\mathscr B$ be the Borel $\sigma$-algebra on $\mathbb R$ and let $\mathscr P$ denote the set of all probability measures on the measurable space $(\mathbb R,\mathscr B)$. Lévy’s metric on ...
9
votes
2answers
755 views

Topology - Quotient Space and Homeomorphism

Consider the topological spaces $A \subseteq X$. Identify the quotient space $X/A$ as a more familiar topological space and prove its homeomorphic. $X = \mathbb{R}$ and $A = \mathbb{Z}$ My thought ...
1
vote
2answers
69 views

Is there another way to describe the category $\mathbf{Top}$ of topological spaces?

The category $\mathbf{Top}_\ast$ of pointed topological spaces can be viewed as the comma category $(\bullet\downarrow\mathbf{Top})$. The objects of the category $\mathbf{Top}$ are topological spaces ...
3
votes
1answer
39 views
+50

Prove that $W \cup S^1$ is connected in the subspace topology of $\mathbb{R^2}$

I want to solve the following question: Prove that the union of $W$ and the unit circle $S^1$ is connected in the subspace topology of $\mathbb{R^2}$ where $W=\{(x, y) \in \mathbb{R^2} | ...
0
votes
2answers
26 views

Show that the Hausdorff topology is unique in a finite set $X$

It's these type of questions that I don't know what approach to even take. Which is stressful in an exam condition. My first thought was, well, why not assume there are at least two different ...
0
votes
1answer
29 views

Why $I \times I $ is not convex in $\mathbb R \times \mathbb R$?

The followings are from Munkers' Topology: and Although I didn't understand Theorem 16.4 but I totally understand the Example 3 above and I believe it's correct. The problem is that $I \times I ...
2
votes
1answer
592 views

Showing that metric induces single unique topology on a finite set

I am trying to prove, that given a metric on a finite set it induces exactly one topology. I have an idea which might lead to a proof, but am not sure: For a finite set X with a given metric d we can ...
0
votes
0answers
14 views

Prove that a finite set $X$ has exactly one topology that arises from a metric $d$

Given $(X,d)$, I need to show the above statement. I found a question here that I initially thought would answer my query Showing that metric induces single unique topology on a finite set However, ...
-1
votes
0answers
17 views

Prove that n-dimensional hyper-rectangle is an open/closed set.

How can I prove that the n-dimensional hyper-rectangle $S=(a_1,b_1)\times (a_2,b_2)\times \cdot\cdot\cdot \times (a_n,b_n)$ is an open set in $\Bbb R^n$ and $T=[a_1,b_1]\times [a_2,b_2]\times ...
1
vote
1answer
55 views

Is $\beta\omega$ hereditarily irresolvable?

$X$ is called resolvable if it can be represented as a union of two disjoint dense sets, it is irresolvable otherwise. Moreover it is hereditarily irresolvable (HI) if every subspace of X is ...
4
votes
1answer
49 views

Property similar to connectedness

Recall that $X$ is connected if $X$ cannot be written as the union of nonempty open sets with empty intersection. Consider the following similar property: $X$ is good if $X$ cannot be written as ...
-1
votes
0answers
9 views

A question about disconnecting finite dimensional Euclidean spaces

Let E be a finite dimensional Euclidean space. If C is a closed subset of E that disconnects E, is it always true that some component of C also disconnects E?
2
votes
1answer
405 views

Discrete metric, singleton open or closed set?

Could someone check the following, is my reasoning correct? EDIT: the following contains errors: see comments Let $$d_\text{disc}(x,y) = \begin{cases}1 & \text{if } x\not = y\\ 0 & ...
1
vote
2answers
33 views

Construct two non-homeomorhic non-Hausdorff topologies on $\{a,b,c\}$

Homeomorphism, the idea and definitions are not too hard to memorize but specific construction/proving it doesn't exist is almost like a millenium question to me. Say in this case; I'm sure my ...
11
votes
2answers
134 views
+50

Why this set is dense in $C_0(\mathbb{R})$

Let $C_0=\{f~|~ f:\mathbb{R}\to\mathbb{R},f~is~continous,\lim\limits_{\vert x\vert \to\infty}f(x)=0\}$ $A=\{f~|~f(x)=p(x)e^{-x^2},p(x)~is~polynomials\}$ Why $A$ is dense in $C_0$. The topology ...
7
votes
1answer
186 views

Continuous path inside the Mandelbrot set connecting i to zero?

This relates to another challenge Question about drawing Mandelbrot filaments. It is possible to compute a formula for a continuous path inside the Mandelbrot Set connecting $c=i$ to $c=0$? ...
1
vote
2answers
27 views

Limit Points of the Empty (Null) Set

I am trying to understand whether or not the statement $X′ ∩ Y′ ⊆ (X ∩ Y)′$ is true or false. In trying to develop a counterexample I started asking myself what sets might make it false. That led ...
1
vote
2answers
33 views

A dense set in $\mathbb{T}$

Let $$\mathbb{T}=\{ z \in \mathbb{C}: |z|=1\} $$ Consider $\mathbb{T}$ as a topological group under multiplication with it's usual topology, I'm reading a proof wich states that a dense set in $T$ ...
4
votes
1answer
42 views

How to show $\mathbb{R}^2/\mathbb{Z}^2$ is homeomorphic to $\mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$

I know that $\mathbb{R}^2/\mathbb{Z}^2$ is homeomorphic to $S_1 \times S_1$ and $\mathbb{R}/\mathbb{Z}$ is homeomorphic to $S_1$ thus the product is homeomorphic to $S_1 \times S_1$. But I wonder if ...
0
votes
2answers
21 views

How can I show that there cannot exist a homeomorphism from $\mathcal{C_1}$ to $\mathcal{C_2}$?

Consider the diagram below with an annulus $\mathcal{A}$ and two circles in the annulus $\mathcal{C}_1$ and $\mathcal{C}_2$. In $\mathbb{R^n}$, any two circles are homeomorphic. But if we have this ...
1
vote
1answer
15 views

Prove that $X$ is regular and second countable iff $X$ is a separable metrizable space

Suppose $X$ is $T_1$ space. Prove that $X$ is regular and second countable iff $X$ is a separable metrizable space. Proof: $\rightarrow$ since $X$ is second countable that implies that $X$ is ...
0
votes
1answer
23 views

What does path-connectedness of $I$ have to do with this at all?

I am utterly confused. Q. Show that $X=\{0,1\}$ with the discrete toplogy is not contractible. Well i need to show that $X$ isn't homotopy equivalent to $\{0\}$. My argument is this We ...
1
vote
0answers
37 views

Proof for Urysohn's lemma.

I've read the proof of Urysohn's lemma from JAMES R. MUNKRES' text.I got it upto very extent.But,I'm not getting intuition for this proof.I liked the proof Proving that a compact subset of a Hausdorff ...
0
votes
0answers
25 views
+50

Equivalent condition for continuity of a bundle map

Let $U$ be a topological space, and let $U \times \mathbb{R}^n$ be the trivial bundle. Let $\varphi: U \times \mathbb{R}^n \to U \times \mathbb{R}^n$ be a bundle map, but assume we don't know whether ...
1
vote
1answer
32 views

Three notions of compactness: minimal conditions for equivalence?

There exist 3 notions of compactness: $X$ is compact if any open cover admits a finite subcover; $X$ is sequentially compact if any sequence in $X$ has a convergent subsequence; $X$ is limit point ...
1
vote
1answer
21 views

How do I prove there exists a such measurable function?

Let $V$ be a closed subspace of $L^2([-1/2,1/2]^n)$. Let $f:(S^1)^n\rightarrow \mathbb{C}$ be a continuous function. Define $\rho:[-1/2,1/2]^n\rightarrow (S^1)^n:x\mapsto (e^{2\pi i x_1},...,e^{2\pi ...
1
vote
2answers
26 views

Prove all closed subspace of a compact space are compact: Redundancy?

I see a redundancy in the following proof of the statement. First, we have a lemma that this proof uses A subspace $A \subseteq X$ is compact if and only if every open cover of $A$ by open subsets ...
2
votes
0answers
16 views

Showing these two selection principles are equivalent.

I stumbled across a few articles about Selection Principles recently, and I'm slowly getting to know more about them as I go along. There's some background before I address the problem I ran into. $X$ ...
0
votes
0answers
15 views

Triangulate this identification space

$Z$ is an identification space of the unit square $Q=\{(x, y) | 0 \leq x, y \leq 1\}$ with the following identifications: $(0, y)$~$(1, y)$ $(x, 0)$~$(x+ \frac{1}{2}, 0) $ $(x, 1) $~ $(x + ...
63
votes
1answer
2k views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
1
vote
1answer
31 views

numerical values of points in cantor set

Let $C$ be the standard middle thirds Cantor set in the interval $[0,1]$. The "endpoints" of $C$ have very simple numerical values that can be listed off: ...
0
votes
1answer
27 views

Let $E$ be a bounded subset of $\mathbb R$, and let $S:=\sup (E)$ be the least upper bound of E.

Let $E$ be a bounded subset of $\mathbb R$, and let $S:=\sup (E)$ be the least upper bound of E. Note that $S$ is a real number from the least upper bound principle. Show that $S$ is an adherent point ...
1
vote
2answers
29 views

Homeomorphism from real interval to an arc of a circle

I haven't seen this question anywhere, surprisingly. In a proof of some theorem, my lecture note abruptly states the above. That Since there is a homeomorphism of any real interval and the arc of ...
2
votes
2answers
411 views

Is this notation for the set of limit points a standard notation?

Well, this doubt is probably silly. We have a standard notation for closure of a set $E$, we denote it $\bar{E}$ or $\operatorname{cl}{E}$ and we have a notation for the interior of a set $E$ we ...
1
vote
1answer
46 views

Cantor-set, product topology

We provide $\{0,1\}$ with the discrete topology and view the product space $X:=\{0,1\}^{\mathbb{N}}:=\prod_{i\in\mathbb{N}}\{0,1\}$ (the space of every 0-1-sequence), with $0\notin\mathbb{N}$ . Show ...
0
votes
1answer
35 views

If $f: \mathbb{R} \to \mathbb{R}\times\mathbb{Q}$ is a continuous function, then $f(\mathbb{R})$ has empty interior.

I am supposed to use connectedness here. Clearly, $f(\mathbb{R})$ is connected. But what now? Thank you.
0
votes
1answer
15 views

Partial converses to extreme value theorem

Under what conditions can we establish a converse to the extreme value theorem? That is, for what topological spaces $(X, \tau)$ can we say that if $(\forall f \in C(X))(\exists c \in E) \left( f(c) = ...
1
vote
0answers
28 views

Let $X=\{0,1\}$ be equipped with the indiscrete topology; Why is every $f:Y \to X$ continuous? [duplicate]

By continuity of $f$, I understand that $f^{-1}X$ must be open in $Y$. Well, the statement is general, i.e. for any space $Y$. Don't know what's in it, don't know what topology it has. Regardless, ...
1
vote
1answer
27 views

Connected subsets of $\mathbb{R}$.

Let $X \subset \mathbb{R}$ be equipped with the subspace topology $T$ with respect to the standard topology on $\mathbb{R}$. When is $X$ connected? It's quite easy to see that if $X$ is an open ...
0
votes
0answers
41 views

Some examples of "Clean topological spaces'

What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X)$ of all real (or complex) valued continuous functions on $X$ is a clean ring. A clean ring is a ...
2
votes
2answers
40 views

Triangulation of torus - understanding why

Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider simplicial complexes. I do not understand why the triangles ...
0
votes
0answers
27 views

prove or give a counter example for the following question. [on hold]

Suppose X is a metric space which is not complete.Then C(X) is complete or not? Prove or give a counter example.
1
vote
0answers
14 views

The existence of homotopy

Is every pair of continuous functions $f:[0,2\pi]\to\Bbb R$ and $g:[0,2\pi]\to\Bbb R$ homotopic? I mean the straight line homotopy $F:[0,2\pi]\times[0,1]\to\Bbb R$ defined by $F(\theta,t)=(1-t)\times ...
1
vote
1answer
25 views

Show that $d( \; \cdot \; ,A)$ 1-Lipschitz continuous

Let $(X, d)$ be a metric space and $A\subset X$, with $$d(x,A) = \inf_{y \in A} d(x,y)$$ Now my problem is to show the following:$$ \forall_{x,z \in X}\mid d(x,A)-d(z,A)| \le |x-z| \;\; \text{(which ...