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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Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...
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96 views

Help understanding $T_1$ & $T_2$ spaces.

What are $T_1$ & $T_2$ spaces? I'm having trouble understanding what exactly are $T_1$ & $T_2$ spaces.
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77 views

Homeomorphism on the Hilbert space

We can consider two different topologies on the Hilbert space ; $l^{2}(\mathbb{N})$. One is the topology deduced from the norm \begin{equation*} \|f\|=\sqrt{\sum_{n=1}^{\infty} f(n)^{2}}, ...
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332 views

How to determine the non-wandering set $\Omega(T)$ (if possible at all)?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and let $T\colon X\to X$ describe the following dynamics: 1 becomes a 2, 2 becomes a 0 and 0 becomes a 1 if at least one of its two neighbours is 1, ...
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36 views

Topologies on a finite set. An open problem?

Some time ago an eminent professor told me about an OPEN problem: Number of possible topologies on a finite set? I was excited about the idea of solving this problem but could not. This was more ...
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18 views

Example of non-homeomorphic compact spaces $K_1$ and $K_2$ such that $K_1\oplus K_1$ is homeomorphic to $K_2\oplus K_2$

Once I heard that there exists two compact spaces $K_1$ and $K_2$ which are non-homeomorphic, but with $K_1\oplus K_1$ homeomorphic to $K_2\oplus K_2$ (where $\oplus$ denotes the topological sum). Is ...
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1answer
24 views

Simples curves on $RP^2$

A subset $\Sigma $ of a space is a simple closed curve if it is homeomorphic to S1. Let $p: S^2 \rightarrow RP^2$ be the canonical projection of the sphere onto the projective plane. Prove that if ...
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57 views

Prove that $\partial A$ is a cutset of connected $X$ if $\operatorname{Int}(A)$ and $\operatorname{Int}(X - A)$ are nonempty

Exercise 6.23 (p.202) of Introduction to Topology: Pure and Applied by C Adams and R Franzosa asks: Let $X$ be a connected topological space and $A$ be a subset of $X$. Prove that if ...
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1answer
59 views

Show that $\mathbb{R}^m$ is not homeomorphic to $\mathbb{R}^n$

Show that $\mathbb{R}^m$ is not homeomorphic to $\mathbb{R}^n$ if $m\ne n$. You may assume that $S^m$ and $S^n$ are different homotopy type if $m\ne n$. My attempt: Suppose $\mathbb{R}^m$ is ...
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27 views

Are these two definitions of basis equivalent?

Lecture note definition Let $(X, \mathcal{T})$ be topological space, A $basis$ of $\mathcal{T}$ is a collection $\mathcal{B}$ of open sets satisfying the following: For each open set $U$ and ...
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22 views

$X,Y$ are compact Hausdorff. $f$ is bijective continuous. Is $f$ a homeomorphism?

Let $X,Y$ be compact Hausdorff spaces. Let $f:X\to Y$ be one-to-one, onto and continuous. Show that $f$ is a homeomorphism. I came up with this "proof" but I am very sure it is wrong. In order to ...
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19 views

construct an example of not desne sets

Construct an example of a topological space $X$ and $A \subset X$ where $int(A) \cup int(X \setminus A)$ is not dense in $X$ which example?
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1answer
8 views

Every locally finite family of non-empty subsets of a Lindelöf space is countable.

I just don't understand the conclusion of the lemma: $|\mathcal{A}| \le \aleph_0$. I think it's related with the fact that every member of $\mathcal{U}$ meets only finitely many members of ...
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113 views

Cancellation in topological product

I was wondering whether $M\times \mathbb{R}$ is homeomorphic to $N\times \mathbb{R}$ implies $M$ is homeomorphic to $N$, where let us say $M,N$ are smooth manifolds. (They are certainly homotopy ...
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18 views

Zips and Zippers

I'm currently reading Differential Manifolds by Antoni Kosinski, and the concept of a zip--defined as half of a zipper--is mentioned early on, of course with the intent of connecting manifolds. This ...
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1answer
34 views

Countable and not closed subset of infinite compact space

The taks is: Show that in every infinite compact space there is a countable subset that is not closed. At first I read that it should be closed and I had an idea to take a point $x_1 \in X$ and an ...
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60 views

Does the product functor preserve quotient maps?

In Hatcher's Algebraic Topology, he presents a proof that if $(X,A)$ satisfies the homotopy extension property, and $A$ is contractible, then $X \simeq X/A$. Part of Hatcher's proof goes: Suppose ...
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1answer
14 views

Paracompact topological space: Why is $\overline{V}_s=F_s$?

Question: Why do they say in the remark that $\overline{V}_s=F_s$? Attempt: The only explanation I can think of is that the union $V_s=\cup_{s(t)=t} A_t$ is finite, and I tried to prove it using ...
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14 views

Closed and boundary subsets

Let $X$ be a nonempty compact space and let $F_1, F_2, ...$be its closed and boundary subsets. Prove that $\bigcup_{n=1}^{\infty} F_n \neq X$ I have no idea how to do it. My only plan would be to ...
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1answer
17 views

The Intersection of Equivalence Relations which cover a relation

Exercise A.3 From John Lee( Topological Manifolds) Let $R \subset X \times X$ be any relation on $X$, and define ~ to be the intersecction of all equivalence relations in $X \times X$ that contain ...
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1answer
15 views

interior, exterior and boundary

Prove $b(int(A)) \subset b(A) $ where $b$ is boundary, $int$ is interior and $ext$ is exterior if $x \notin b(A)$ then $ x \in int(A) \cup ext(A) $ if $x \in int(A) \to x \in int(int(A))$ ...
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81 views

Hatcher 2.1.10…

Hatcher asked a question in chapter $2$ (a) Show the quotient space of a finite collection of disjoint $2$-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic ...
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1answer
37 views

Non-Empty Finite Subset $U$ of $\mathbb{R}$ is not Open

Consider $(\mathbb{R}, \mathcal{T})$ standard topology Definition : $ U \in \mathcal{T}$ if $\forall x \in U, \exists \delta$ such that $(x-\delta,x+\delta) \subset U$ If using this definition, ...
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25 views

Connected set in normed space

I have this exercise: "let $E$ be a normed space and $X\subset E$ $$X~\text{connected}~\Longleftrightarrow \forall A\subset X,~\text{such that} A\neq\emptyset, A\neq X~\text{we have}~ Fr(A)\neq ...
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90 views

What do Set-Theoretic (General) Topologists study? [on hold]

I was reading in Elementary Topology by O Viro, O Ivanov, V Kharlamov, and N Netsvetaev and it caught my attention the following quotes by the authors: "...As a research field (refering to General ...
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1answer
40 views

Angle form, 1-form, proof verification.

Check that the $1$-form $d\,\text{arg}$ in $\mathbb{R}^2 - \{0\}$ is just the form$${{-y}\over{x^2 + y^2}}\,dx + {{x}\over{x^2 + y^2}}\,dy.$$ My solution is as follows. Observe that we can ...
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16 views

If $D$ from $X*X $ to $R$ with this condition that $ D(x,y)=-D(y,x)$, and if $ D(x,y)\ge0$, $D(y,z)\ge0$, can we implies that $D(x,z)\ge0$? [on hold]

If there is a function $D$ from $X*X $ to $R$ with this condition that $ D(x,y)=-D(y,x)$, and if $ D(x,y)\ge 0$, $D(y,z)\ge0$, can we implies that $D(x,z)\ge 0$?
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34 views

Meaning of n-connected pairs

A topological space $X$ is $n$-connected if the homotopy groups $\pi_r(X)$ for $0 \leq r \leq n$ are trivial groups. This means (let's say geometrically), $X$ is $0$-connected if it is non-empty and ...
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48 views

A function continuous on rational points and discontinuous on irrational points

How to find function $f : \Bbb R \to \Bbb R$ such that $f$ is continuous on the rational numbers and discontinuous at irrational numbers? I was told to use the Baire Theorem to show that the set of ...
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5k views

Projection map being a closed map

Let $\pi: X \times Y \to X$ be a projection map where $Y$ is compact. Prove that $\pi$ is a closed map. First I would like to see a proof of this claim. I want to know that here why compactness is ...
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26 views

Interior, closure, isolated points and boundary of a set of a normed vector space

Let $X =(\mathbb{R}^2,||(x_1,x_2)|| := |x_1| +|x_2|)$ be a normed vector space. Find the interior, closure,Isolated points, and boundary of $Y =\{(x, \frac{1}{n})~|~ x\in \mathbb{R} \wedge n\in ...
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1answer
18 views

Filter of sets containing a subset converges

I'm just learning about filters, and I came across the following exercise in Willard's Topology: Let $X$ be a topological space and $A \subset X$. The cluster points of the filter $\mathcal{F} ...
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42 views

Closed set, closure of a set

Prove if A is open then $A \cap \bar{B} \subset \overline{A \cap B}$ $ A \cap \bar{B}= A \cap (B \cup B')=(A \cap B) \cup (A \cap B')$ $A \cap B \subset \overline{A \cap B} $ then I have to ...
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2answers
48 views

minimal embeddings of topological spaces into connected spaces

Defintions: Let $X$ be a topological space. 1) A connected space $Y$ is a minimal connected ambient (m.c.a for short) space for $X$ if there exists an embedding $i:X\mapsto Y$, and for every ...
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1answer
86 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
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1answer
43 views

Does map induced by rotation preserve the volume form?

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a rotation. My question is, does the map of $S^{n-1}$ onto $S^{n-1}$ induced by $A$ necessarily preserve the volume form?
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19 views

G-P Exercise 4.8.2, proof verification.

Let $\gamma$ be a smooth closed curve in $\mathbb{R}^2 - \{0\}$ and $\omega$ any closed $1$-form on $\mathbb{R}^2 - \{0\}$. Prove that$$\oint_\gamma \omega = W(\gamma, 0) \int_{S^1} \omega,$$where ...
4
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1answer
65 views

Questions about a topological proof of the FTA

I'm a high school student, curious about proofs of the Fundamental Theorem of Algebra. Specifically, I've been thinking about one of the topological proofs of the theorem, given in Courant's book, ...
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2answers
86 views

Homeomorphism between $\prod X_i$ and the Cantor set

I've been trying to find a homeomorphism $$\Phi:\prod_{i=1}^\infty X_i \to \{ 0, 1\}^\mathbb{N},$$ where each $X_i$ is a finite set with at least two elements, but have been unable to. Things I've ...
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114 views

The Cantor set is not strong measure zero

$A \subseteq \mathbb R$ is strong measure zero if given any sequence $( \epsilon_n )_{n \in \mathbb N}$ of positive reals there is a sequence $( I_n )_{n \in \mathbb N}$ of open intervals such that ...
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1answer
24 views

Orthogonal group acts on vector field

I recently had an exam, yesterday acctually, and there was a question that stumped me. The orthogonal group $O(n)$ acts on $\mathbb{R}^n$ by matrix multiplication, show that the orbit space is ...
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30 views

Can't work out if this proof is sound or not. Any ideas?

Let $V$ be a normed space over some field $\mathbb K$. I proved that $$ \overline{B_r(a)} = \{v \in V \mid \|v-a\| \le r \}$$ $\subseteq $ was easy but for the $\supseteq$ direction I am really not ...
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1answer
387 views

A question about the contractibility of the Sierpinski space

The two-point Sierpinski space is usually defined as follows: Let $X =\{x,y\}$ be the two-point space where the only open sets are $X, \varnothing, \{x\}$. I think from this it can be inferred that ...
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35 views

Help in understanding a conjugacy problem

I am studying the book Applied Symbolic Dynamics and Chaos By Bai-lin Hao, Wei-Mou Zheng The basic premise of the concept of Symbolic Dynamics is : "Symbolic ...
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2answers
17 views

Convex Homotopy

Suppose $f , g : X \to U \subset \mathbb R^2$ are two mappings from a topological space $X$ to a convex set $U$. Prove that $f$ and $g$ are homotopic, using only the definition of the product ...
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31 views

Is the intersection of two locally compact locally compact?

Taking locally compact as such that every point has a local base of compact neighborhoods, is the intersection of two locally compact subspaces locally compact?
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12k views

Why are there 12 pentagons and 20 hexagons on a soccer ball?

Edge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron. Is there some insight into why the alternation of pentagons and hexagons yields an approximated ...
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24 views

a set $S \subseteq \mathbb{R}$ is closed and bounded if and only if every sequence in $S$ has a sub sequence converging to a point of $S$

Prove that a set $S \subseteq \mathbb{R}$ is closed and bounded if and only if every sequence in $S$ has a sub sequence converging to a point of $S$. The direction $\Rightarrow$ was easy. But I don't ...
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1answer
16 views

How to prove that every Paracompact space with the Suslin property is Lindelof

This question was asked a few years ago and a proof was given here http://math.stackexchange.com/a/190147/235467. However, in this proof it states that paracompactness implies the existence of a ...
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32 views

Does a homogeneous metrizable space admit a compatible homogeneous metric?

Assume that X is a compact metrizable topological space for which the action of homeomorphism group is transitive. Is there a compatible metric d on X such that the action of group of isometries ...