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
44 views

Misunderstanding of $\epsilon$-neighborhood

I am given the following definition of an $\epsilon$-neighborhood: Given a real number $a\in \mathbb{R}$ and a positive number $\epsilon>0$, the set $$\{x \in \mathbb{R}: |x-a|<\epsilon \}$$ ...
1
vote
0answers
7 views

Topologies conceptual confusion (topology of maximum norm/of pointwise convergence)

One of the questions from my lectures notes reads as follows: "Show that the identity map from $C[0,1]$ with the topology, $T_m$, induced by the maximum norm to the topology of pointwise convergence, ...
-1
votes
0answers
34 views

Existence of two-point sets

How to prove the existence of the two-point sets? I understand that demonstrated using axiom of choice but do not understand well as it does. Thanks.
0
votes
1answer
18 views

Connectedness of a cartesian product

Let $X,Y$ be topological spaces. Show that the productspace $X\times Y$ is connected $\Leftrightarrow X,Y$ are connected. Could someone give me some pointers? I'm not looking for a full solution, ...
1
vote
0answers
27 views

technique of showing a set is a topology

Let $X $ be a set and let $Y\subset X$. Define $\tau_Y $ to be the collection of all subsets U of X such that $Y\subset U $ or $U= \emptyset $ . Prove that $\tau_Y $ is a topology on $X $. I have ...
1
vote
1answer
41 views

$X_1,X_2$ connected. Show that $X_1\times X_2$ is connected.

If $X_1,...,X_n$ are connected topological space, show that $X=X_1\times ...\times X_n$ is also connected. I have the proof in front of me but I couldn't understand. It starts by claiming that: ...
-8
votes
0answers
38 views

please help me this is about my final exam [on hold]

show that in a Hausdorff space with a countable basis all four varieties of compactness(compactness, limit point compactness,sequential compactness, countable compactness) are equivalent
0
votes
0answers
6 views

Looking for a general point that can be found in the highest dimensional irreducible component of the $k$-secant variety.

Let $X\subseteq\mathbb{P}^{n}$ be an irreducible variety, $p_{1},\ldots,p_{k}\in X$ general points and $p\in\langle p_{1},\ldots,p_{k}\rangle$ a general point. I want to show that we can consider $p$ ...
2
votes
0answers
31 views

About contractible implies simply connected [duplicate]

most books say that contractible implies simply connected is trivial as at time one, every point of the space is shrunk to one point, including the loop. However, there is a problem. In discussing ...
2
votes
1answer
46 views

Non-continous topology?

I've been studying topology this term and it really got me interested. But sometimes in math I feel that we are just taught things one by one, without really talking about why we do it that way. So I ...
2
votes
1answer
14 views

Decomposition into irreducibles of a Noetherian topological space

I'm struggling with the proof that says a Noetherian topological space $X$ is the finite union of closed irreducible subsets. In particular with this part: First observe that every nonempty set of ...
0
votes
1answer
26 views

if two space are homotopy equivalent and one is connected, prove that the other is connected as well

I've tried using the definition of homotopy equivalent spaces which states that X and Y are homotopy equivalent if: There are continuous functions $f:X \rightarrow Y,g:Y \rightarrow X$ such that $f ...
1
vote
0answers
8 views

Are there any other measures of complexity for a continuous map than topological entropy?

Let $X$ be a compact topological space and $T\colon X\to X$ be continuous. In order to say something about the complexity of $(X,T)$ there is of course the notion of topological entropy of $T$, ...
-1
votes
0answers
9 views

Forms on Varieties [on hold]

Given M n-dimensional differentiable variety, orientable and compact, and $w_0$ an orientation form of M, we want to proof: i) Given $\rho \in \Omega^r(M)$, $0 \leq r \leq n$ and $g: W \rightarrow M$ ...
1
vote
1answer
23 views

Categorical version of the Tietze Extension Theorem

In Donald Hartig's short paper An Important Functor in Analysis and Topology, Theorem 1 is preceded by the following statement: Since the spaces we are dealing with are compact, a one-to-one map ...
2
votes
2answers
22 views

Existence of a countable basis in the definition of a manifold and uncountable bases.

In the definition of a manifold $M$ of dimension $n$ in An Introduction to Differentiable Manifolds and Riemannian Geometry by William M. Boothby (page 6), the third criterion is $M$ has a ...
0
votes
4answers
47 views

How does this specific set look like?

How does the set $ \begin{pmatrix} \cos(x) \\ 4\sin(x) \\ \end{pmatrix} \in \mathbb R ^2$ with $x \in \mathbb R$ look like? I guess it should be similar to $f(x) = ...
2
votes
1answer
20 views

Characterize R0-space by convergent filters

I want to prove the equivalence of the two following characterizations of R0-spaces. One comes from my textbook (with filters), the other one is taken from wikipedia. First, I will introduce the ...
5
votes
1answer
53 views

Characterization of 1-dimensional manifolds. [duplicate]

My intuition tells me that the only connected 1-dimensional topological manifolds are the real line $\mathbb{R}$ and the circle $S^1$. Is this true? If yes, is it possible to prove it from first ...
1
vote
0answers
15 views

Upper-hemicontinuity of product maps on compact metric spaces.

Let $X$ and $\{Y_i\}_{i\in I}$ be compact metric spaces (where $I$ an index set of possibly uncountable cardinality). Let $\Gamma_i$ be a compact valued, upper hemicontinuous (UHC) correspondence from ...
1
vote
1answer
19 views

Why boundary of a locally closed set is nowhere dense?

Let $X$ is locally closed , i.e. exist open $U$ S.t. $X=\overline{X} \cap U $ , and $bd (X) = \overline{X} \setminus \mathring{X} $. How can I show that $ bd(X) $ is nowhere dense? I read topics ...
4
votes
5answers
88 views

Why is $\operatorname{Int}(A) \cup \operatorname{Int}(B) \neq \operatorname{Int}(A \cup B)$?

I know that $\operatorname{Int}(A) \cup \operatorname{Int}(B) \subset \operatorname{Int}(A \cup B)$, but that the other direction does not hold, so can anybody please tell me whats wrong with the ...
1
vote
3answers
29 views

How continuity of $f$ and path-connectedness of $g$ results in $f\circ g$ to be path-connected?

Theorem 6.29 (p.213) of Introduction to Topology: Pure and Applied by C Adams and R Franzosa says: Assume that $f : X \rightarrow Y$ is continuous and $X$ is path connected. Then $f (X)$ is a ...
1
vote
0answers
33 views

Maps between groups and classifying spaces

Suppose we have two Lie groups $G$ and $H$, as well as two homomorphisms $\phi_1,\phi_2 \colon G \to H$ and an arbitary continuous map $g \colon G \to G$. Futhermore suppose that $\phi_2$ is homotopic ...
5
votes
1answer
45 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 ...
2
votes
2answers
54 views

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 ...
0
votes
1answer
51 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 ...
4
votes
1answer
42 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 ...
0
votes
2answers
32 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 ...
-1
votes
2answers
40 views

construct an example of not dense sets [on hold]

Construct an example of a topological space $X$ and $A \subset X$ where $\operatorname{int}(A) \cup \operatorname{int}(X \setminus A)$ is not dense in $X$ which example?
2
votes
1answer
14 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 ...
3
votes
1answer
68 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 ...
1
vote
0answers
25 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 ...
1
vote
1answer
23 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 ...
0
votes
1answer
15 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 ...
1
vote
1answer
43 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 ...
1
vote
1answer
21 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 ...
1
vote
1answer
22 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 $x \in int(A) \to x \in int(int(A))$ ...
2
votes
0answers
26 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 ...
1
vote
1answer
42 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, ...
-3
votes
0answers
18 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 $D(x,z)\ge 0$?
3
votes
0answers
100 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 ...
0
votes
0answers
52 views

A function continuous on rational points and discontinuous on irrational points [duplicate]

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 ...
2
votes
1answer
46 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 ...
1
vote
2answers
28 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 ...
2
votes
2answers
38 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 ...
4
votes
0answers
26 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 ...
2
votes
2answers
51 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 ...
3
votes
1answer
25 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} ...
0
votes
1answer
25 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 ...