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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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 ...
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0answers
19 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 ...
4
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0answers
22 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
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2answers
47 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 ...
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1answer
32 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
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1answer
37 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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2answers
25 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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2answers
34 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?
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1answer
9 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
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1answer
64 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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0answers
20 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
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0answers
17 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
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 ...
1
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1answer
37 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
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1answer
19 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
18 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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0answers
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 ...
1
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1answer
39 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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0answers
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 $D(x,z)\ge 0$?
3
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0answers
95 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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0answers
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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1answer
40 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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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
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2answers
35 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
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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
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2answers
49 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
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1answer
22 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
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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 ...
3
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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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1answer
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 ...
4
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1answer
65 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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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
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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 ...
0
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2answers
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 ...
1
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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 ...
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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
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0answers
34 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
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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: $(\Rightarrow)$ ...
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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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4answers
34 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 ...
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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 $x_n\to x_0\in\mathbb{R}^n \Rightarrow x_0\in M$? ...
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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
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1answer
39 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
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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 ...
2
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0answers
36 views

Tetrahedron and balls in space

A right tetrahedron and a ball arbitrarily located in space are given. It is allowed to reflect the tetrahedron from each of its faces. It is possible to place the center of the tetrahedron inside the ...
0
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1answer
26 views

Topology on $[X]^2$ for Hausdorff space $X$

Let $(X,\tau)$ be a Hausdorff space. Let $[X]^2 = \big\{\{x,y\}: x,y\in X \land x\neq y\big\}$. For $U,V\in \tau$ with $U\cap V = \emptyset$ we set $[U,V] = \big\{\{x,y\} \in [X]^2: x\in U\land y\in ...
2
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1answer
20 views

Proof product of components in factors is a component in product topology

Let $x = (x_1, x_2, .... x_{n})$ be a point in a product space $(Y, \tau_{Y}) = \prod_{i = 1}^{n} (X_{i}, \tau_{i})$. The component $C_{X}(y)$ in a topological space is the union of all connected ...
0
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1answer
43 views

If there is a finite-closed topology on $X$ with 3 clopen elements, then $X$ is finite

Let $T$ be a finite-closed topology on $X$. $X$ has 3 clopen elements. Prove that $X$ is finite. Empty set must be one of these clopen sets as well as $X$. Therefore, we are left with some element ...
1
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1answer
67 views

Covering spaces of $S^1$

Put $\tilde X=\lbrace (exp(2\pi if(t)),t)| t\in \mathbb{R} \rbrace$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ is any continuous function and let $\pi_1$ be the projecction on the first coordinate. ...
2
votes
3answers
51 views

(Non-Euclidean) Compactness

Compactness in Euclidean Space The only definition of compact set that ever made sense to me was the intro calculus one: A set is called compact if it is closed and bounded. ...