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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9
votes
1answer
44 views

Can the product of two $\mathsf Y$'s be embedded in 3-space?

Let $Y$ denote the space homeomorphic to the (sans serif) letter $${\huge\mathsf Y}$$ or, equivalently, the space of three closed intervals glued together at one endpoint. Consider the space $Y\times ...
0
votes
0answers
11 views

Visualization: Homeomorphism from a Lattice Quotient to a Torus

I find that sometimes in mathematics one tends to get bogged down in the technical details and lose sight of what's actually going on. Of course this opinion is coming from my limited experience as an ...
1
vote
1answer
15 views

Every open cover of a smooth Manifold has a regular refinement

I am trying to understand the proof of Let M be a smooth manifold. Every open cover of M has a regular refinement. The proof begins as follows [Lee] : Let $X$ ...
0
votes
2answers
27 views

continuous function in a topological space

It is known that if $f, g$ are continuous functions then $f+g$ is also continuous. I want to know how to prove it in topological language, thst is, $f$ is continuous if for any $x$ and any open ...
-3
votes
0answers
25 views
0
votes
2answers
35 views

Needing a clearer idea on “quotient topology”

So here's the definition of a quotient topology Let $(X,\tau)$ be a toplogical space and define some equivalence relation $\sim$ on the set $X$. There exists a natural surjection denoted $p:X \to ...
5
votes
2answers
137 views

Given a pair of continuous functions from a topological space to an ordered set, how to prove that this set is closed?

Given that $X$ is an arbitrary topological space, $Y$ is a totally ordered set in the order topology, and $f$, $g \colon X \to Y$ are continuous functions, how to show that the subset $A$ of $X$ given ...
1
vote
1answer
16 views

Connected Components for $(\Bbb R, \mathcal T_{ lower limit})$

$(\Bbb R, \mathcal T_{{ lower }{ limit}})$ is a topological space $\Bbb R$ with Lower limit topology. As I know, $(\Bbb R, \mathcal T_{{ lower }{ limit}})$ is disconnected. What are the Connected ...
1
vote
1answer
76 views

The closure of $\mathbb{N}$ is $\mathbb N$. The closure of $\mathbb Z$ is $\mathbb Z$… etc

Prove this lemma Lemma: The closure of $\mathbb{N}$ is $\mathbb N$. The closure of $\mathbb Z$ is $\mathbb Z$. The closure of $\mathbb Q$ is $\mathbb R$, and the closure of $\mathbb R$ is $\mathbb ...
0
votes
1answer
14 views

What are the interior and limit points of the given subset of n-dimensional Euclidean space?

Considering metric topology and giving the set E subspace topology in the Euclidean Space. Given E={(a$_{1}$, a$_{2}$,...,a$_{n-1}$, 0) | a$_{i}\in\mathbb{R}$} $\subset$$\mathbb{R}^{n}$. I want to ...
0
votes
1answer
27 views

Let $X$ be a subset of $\mathbb R$. Show that $\bar X$ is closed

Let $X$ be a subset of $\mathbb R$. Show that $\bar X$ is closed (i.e, $\bar {\bar X} = \bar X$). Futhermore, show that if $Y$ is any enclosed set that contains $X$, then $Y$ also contains $\bar{X}$. ...
0
votes
1answer
34 views

if there exists a sequence $(a_n)^{\infty}_{n=0}$, consisting entirely of elements in $X$, which converges to $x$.

Prove the Lemma: Let $X$ be a subset of $\mathbb{R}$, and let $x \in \mathbb{R}$. Then $x$ is an adherent point of $X$ if and only if there exists a sequence $(a_n)^{\infty}_{n=0}$, consisting ...
1
vote
0answers
16 views

Prove that the union of two given subsets of $\Bbb{C}^n$ is path-connected

Consider a subset $A$ of $Z=(\Bbb{C}^n$, Zariski topology) and regard it as a subspace of ($\Bbb{C}^n$, Metric topology). Sine $\Bbb{C}^n$ is homeomorphic to $\Bbb{R}^2n$, we can decide if A is ...
0
votes
1answer
4 views

Prove that that a map in $C^N$ in the metric topology is continuous to the Zariski topology, but that the map is not a homeomorphism.

Prove that that a map in $C^N$ in the metric topology is continuous to the Zariski topology, but that the map is not a homeomorphism. My Work: I plan to use the fact that the metric topology is a ...
1
vote
1answer
36 views

Give an example of two subsets $X,Y$ of the real line such that $\overline{X\cap Y}\neq \bar X \cap \bar Y$.

Give an example of two subsets $X,Y$ of the real line such that $\overline{X\cap Y}\neq \bar X \cap \bar Y$. My Attempt $X=(0,1)$ $Y=(1,2)$ Is this right? What can I say about it?
8
votes
5answers
1k views

Is a ball always connected in a connected metric space?

If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?
0
votes
0answers
11 views

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 ...
2
votes
3answers
154 views

Continuous functions in the indiscrete topology?

Slight curiosity. I've learned not to question too much in topology and basically, acquiesce. In a sense, thinking hard or trying to be smart in this area of study is a suicide mission for newbies. ...
1
vote
1answer
22 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 ...
1
vote
2answers
34 views

Closed Euclidean ball in 1D and the Brouwer fixed point theorem

I'm slightly unsure about how the theorem is presented to me in lecture... The Brouwer fixed point theorem for $1$ dimension. Every continuous map $[0,1] \to [0,1]$ has at least one fixed point. ...
0
votes
1answer
21 views

Show that two metrics known not to be strongly equivalent actually induce the same topology.

Suppose on $\mathbb{R}$, we have the usual Euclidean metric, $\rho_{1}(x,y) = \Vert x-y \Vert$, and also the metric $\rho_{2} = \displaystyle \frac{\rho_{1}(x,y)}{1+\rho_{1}}$. I need to show that ...
0
votes
1answer
28 views

Prove that $X$ is finite.

Let $\mathcal T$ be the finite-closed topology on a set $X$. If $\mathcal T$ is also the discrete topology, prove that the set $X$ is finite. My attempt: We know that $\emptyset$ and $X$ are in ...
0
votes
2answers
40 views

Show that a discrete set is at most countable

How to show that a discrete subset $A$ of $\mathbb C$ is at most countable? Ok I read the relevant questions here but I still don't understand how I should create an injection from $A$ to $\mathbb Q$. ...
1
vote
0answers
15 views

Does a certain type of connected subset exist in Euclidean spaces having an arbitrarily high dimension

Let $\mathbb R$ be the set of all real numbers and for each positive integer $n$, let $f_n$ be a mapping of $\mathbb R^n$ into $\mathbb R$. For each positive integer $n$, does there exist an $f_n$ ...
-1
votes
1answer
30 views

How to understand Urysohn lemma and The Tychonoff Theorem . [on hold]

I can not understand Theorem 33.1 (Urysohn lemma) and Theorem 37.3 The Tychonoff Theorem from Topology by Munkers Can anyone told me this lemma in single idea. Please Help me ...
1
vote
2answers
20 views

does the condition “every open set is a countable union of closed sets” imply metrizability

In metric spaces, every open set is a countable union of closed sets. is the converse true? A topological space with the property "every open set is a countable union of closed sets" has to be ...
0
votes
2answers
31 views

Free cocompact action of discrete group gives a covering map

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...
3
votes
0answers
28 views

Intuition for the definition of a dense functor?

A functor $i:S\to C$ is dense if every object $c$ of $C$ is the vertex of the colimit of the following diagram $\varinjlim((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C)$. I ...
1
vote
1answer
27 views

Compact topological Space with Closed Graph implies continuity

Let $X,Y$ be topological spaces and $f:X \rightarrow Y$ If $Y$ is compact and $G(f)$ (graph of $f$) is closed, show that $f$ is continuous. I consider an open set in $X$, and as $G(f)$ is closed, ...
2
votes
1answer
27 views

Why is $(-\infty, \sqrt{2}) \cap \mathbb{Q}$ open in $\mathbb{Q}$?

I am looking at an example of disconnected spaces. Let $U=(-\infty, \sqrt{2})\cap \mathbb{Q}$ and $V=(\sqrt{2},\infty) \cap \mathbb{Q}$. Then $U,V$ are open in $\mathbb{Q}$ by definition of the ...
1
vote
2answers
32 views

Comparing “size” of open sets in a topological space?

In metric space, we have the notion of distance, which enables us to have a general idea on the "size" of an open set, say, by measuring its diameter. If we now have 2 open sets $A,B$ in a topological ...
0
votes
1answer
29 views

Quotient topology on unit sphere

Let $\sim$ be the equivalence relation $$a\sim b\iff a=b\text{ or }a=-b,$$ for $a,b$ on the unit sphere $S^2$. Let $Q$ be the quotient space. How do I show that the quotient map is a covering ...
1
vote
2answers
495 views

continuity in the strong topology implies continuity in the weak one

I have to prove that if $T:(E,\|\cdot\|_E)\rightarrow (F,\|\cdot\|_F)$ is a continuous and linear operator, and $x_h\rightharpoonup x$ in $E$, than $Tx_h\rightharpoonup Tx$ in $F$. So we know that $T$ ...
4
votes
1answer
39 views

Continuous map in $\mathbb{R}^2$ has a (scaled) fixed point

Let $\phi:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a continuous map. How do I prove that there exist $a>0$ and $x\in\mathbb{R}^2$ such that $\phi(x)=ax$? What I know: I thought maybe this can ...
-2
votes
1answer
25 views

Is it true that hausdorff and continuous lead to first separation axiom [on hold]

Suppose that $(X, T_X)$ and $(Y, T_Y)$ are topological spaces and that $f : X \longrightarrow Y$ is a continuous map and $X$ is Hausdorff. Is it true that $Y$ satisfies the first separation axiom?
0
votes
1answer
17 views

Problem computing the fundamental group of the double torus.

To compute this I want to use the Seifert-van Kampen Theorem, then I choose $U=\text{Left Torus}/{point}$ and $V=\text{Right Torus}/{point}$ so then the intersection $U \cap V$ is a cylinder with out ...
0
votes
1answer
27 views

Visualizing the quotient of a torus and a circle

We were asked to compute the homology for the double torus, $X$, and a circle around one of the loops, $B$, of the torus (not a circle between the two halves of the torus) and were told that this ...
0
votes
2answers
31 views

$f$ is null homotopy if and only if $f_*$ is the trivial induced homomorphism

I want to show that $f:S^1\to S^1$ is homotopic to the constant map if and only if $f_*:\pi_1(S^1,x_0)\to \pi_1(S^1,x)$ , $f_*([\gamma])=0$ This seems like it should be an obvious fact but I am having ...
0
votes
1answer
12 views

On $\Bbb R^2$, Are unit circle centred at the origin and the origin homotopic equivalent?

I guess these two spaces are not homotopic equivalent. I suppose there are homotopic equivalent. Let $X=\{x \in \Bbb R^2 : ||x||=1\}$ $Y=\{(0,0)\}$ And there exists two functions $f: X\to \{(0,0)\} ...
1
vote
1answer
36 views

Are 0 and -1 the only rational periodic solutions of $z_{n}\equiv z_{n-1}^{2}+c$?

Let $c$ be any complex rational number. Let $z$ be a series of polynomials in $c$ defined by $z_{n}\equiv z_{n-1}^{2}+c$ and $z_{0}\equiv0$ The only rational roots of any $z_{n}$ I have been able ...
1
vote
1answer
73 views
+50

Do non-second-countable spaces have “small” non-second-countable subspaces?

If $X$ is any space which is not second-countable, can one find a subspace $Y \subseteq X$ with $|Y| \leq \aleph_1$ which is also not second-countable? (Recall that a topological space $X$ is ...
3
votes
0answers
41 views

Where can I learn more about “topological germs”?

Definition 0. By a topological germ, I mean a pointed topological space. Whenever $X$ and $Y$ denote topological germs, by a morphism of topological germs $X \rightarrow Y$, I mean a neighbourhood ...
0
votes
0answers
24 views

Free action of a discrete group gives a covering space

I'd like to find a short proof of the following seemingly basic fact. Suppose a discrete group $G$ acts freely on a manifold $X$ with the quotient $X/G$ being compact. Then $X$ is a covering space of ...
0
votes
1answer
45 views

Showing $\mathbb{R}^3$ minus $n$ parallel lines is homotopic to $\mathbb{R}^2\setminus\{p_1,\dots,p_n\}$

I want to show that if I remove $n$ parallel lines from $\mathbb{R}^3$ then I get $\mathbb{R}^2\setminus \{p_1,\dots,p_n\}.$ There is also some underlying structure I wish to also understand. That ...
0
votes
1answer
51 views

Show that $\{1/n:n∈N\}∪\{0\}$ is compact

The set is in $R^1$ and consists of $0$ and the numbers $1/n$. Call it $E$. Take a set of $n$ intervals of radius $r$, centered less than $2r$ apart and such that $\sum_{i=1}^n r \ge 1/2$. Call the ...
0
votes
2answers
34 views

Two decreasing, convex functions agreeing on a closed set

Fix a closed subset $B$ of $[0,\infty)$ and assume that $0\in B$. I am striving to construct two functions $f,g:[0,\infty)\to\mathbb R$ such that $f(0)=g(0)=1$; $f(x)\geq 0$ and $g(x)\geq0$ for each ...
0
votes
0answers
39 views

Isomorphism theorems for topological groups

I know that the second isomorphism theorem for groups doesn't hold for topological groups, the version that I have for the second isomorphism theorem is: If $G$ is a group, $H$ a subgroup of $G$ and ...
0
votes
1answer
23 views

Induced topologies by Metric Spaces continuous

Let $(X,d)$ be a metric space and $T$ the corresponding topology on $X$. Fix $a$ as an element in X. Prove that the map $f: (X,T) \to \mathbb{R}$ defined by $f(x) = d(a,x)$ is continuous. I know I ...
2
votes
1answer
40 views

The set where a derivative vanishes is G-delta

If $f:I\to R$ ($I$ - interval) is differentiable, then $\{x\colon f'(x)=0\}$ is a $G_{\delta}$ set. The lecturer didn't prove this fact and I found no proof in my books. How it can be proven?
-1
votes
1answer
23 views

Hausdorff space with a additional property

Suppose a Hausdorff topological space $(X, \tau)$ has the following property. A and B are disjoint closed subsets of $X$ implies there is a continuous function say $f_{AB} : X \rightarrow [0,1]$ such ...