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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0
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4answers
44 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
16 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
46 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
13 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
18 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
84 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
28 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
30 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
0answers
28 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
52 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
42 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
40 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
29 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
39 views

construct an example of not dense sets

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
13 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
66 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
24 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
0answers
18 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
42 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
20 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
21 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
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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
40 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
17 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
98 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
51 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 ...
1
vote
1answer
43 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
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 ...
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
20 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
24 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
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 ...
3
votes
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?
0
votes
1answer
31 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 ...
4
votes
1answer
66 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 ...
1
vote
2answers
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?
7
votes
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 ...
0
votes
2answers
44 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 ...
1
vote
2answers
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 ...
-3
votes
1answer
52 views

can anyone help me with following question attached in image file [on hold]

Let $(X,\|\cdot\|)$ be a normed space, where $$X=\{(a_n)_{n\geq 1} \mid (a_n)_{n\geq 1} \text{, bounded real sequence}\}$$ and $$\|(a_n)_n\|=\sup_{n\in N} |a_n|$$ Let $$ M=\{(a_n)_n\in X\mid 0\leq ...
3
votes
0answers
39 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 ...
1
vote
1answer
21 views

Separable iff Lindelof for pseudometric spaces

I'm trying to prove, for $X$ a pseudometric space $$X \text{ Lindelof } \Leftrightarrow X \text{ separable }$$ Here are some of my ideas so far - the forward direction should work: $(\Rightarrow)$ ...
0
votes
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 ...
0
votes
4answers
35 views

$\{-n+\frac{1}{n};n\in\mathbb{N}\}=M$ closed in $\mathbb{R}$

Why is $\{-n+\frac{1}{n};n\in\mathbb{N}\}=M$ closed in $\mathbb{R}$ (here is $\mathbb{R}$ endowed with the standard topology? I could use the criterion: Is $(x_n)\subseteq M$ such that $x_n\to ...
1
vote
1answer
14 views

$\partial M\subset M$ implies (Is $(x_n)\subseteq M$ such that $x_n\to x_0\in\mathbb{R}^n \Rightarrow x_0\in M$)

Let $M\subset \mathbb{R}^n$. I want to how to proof: Why implies 1. $\partial M\subset M$ this type of closedness: 2. Is $(x_n)\subseteq M$ such that $x_n\to x_0\in\mathbb{R}^n \Rightarrow x_0\in M$? ...
-1
votes
0answers
31 views

Set of continuous functions that vanish at infinity is complete

Why is it easy to see that a set of all continuous functions $C_0$ that vanish at infinity implies that each $f\in C_0$ is bounded and the set is complete with respect to the uniform (sup) -norm? ...
2
votes
1answer
40 views

Open and Closed covering

Let $X$ be a compact Hausdorff and totally disconnected space and $A$ be a closed subset of $X$ contained in an open set $U$. Then we can find a finite set $\{V_1,\cdots,V_n\}$, where each $V_i$ is ...
0
votes
2answers
18 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 ...