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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7 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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1answer
14 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
12 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
19 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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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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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
73 views

What do Set-Theoretic (General) Topologists study?

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
41 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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15 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 ...
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2answers
31 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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0answers
17 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
44 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 ...
2
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1answer
15 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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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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1answer
38 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
29 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 ...
3
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1answer
51 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
30 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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1answer
37 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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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 ...
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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
50 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 ...
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0answers
29 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 ...
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1answer
20 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
30 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
38 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 ...
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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
35 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 ...
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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 ...
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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 ...
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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 ...
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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
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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. ...
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1answer
53 views

The intersection of dense subset and open subset

Let $A$ be a dense subset of $X$, and $B$ let be a non-empty open subset of $X$. Prove that $A\cap B \not = \emptyset $. if A is dense in X then $ \bar{A}=X=A\cup A'$ where $A'$ is the derived set ...
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18 views

Relationships Between Moduli Space and Objects They Parametrize

My friend and I were wondering recently what, if any, are the relationships between the geometric properties of a moduli space and the geometry of the objects that the space parametrizes. As an ...
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1answer
43 views

Question on one point compactification

I was given the following question in my general topology class assignment which is multi parts - most of which I managed alright by myself some of which I need help on. We are given a non compact ...
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1answer
96 views

Can something contain iteself? [on hold]

I asked this over on the Phyisics part of StackExchange, and they suggested I move my question here. And said question is: Can something contain itself? The question is simple enough, and I can ...
3
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4answers
94 views

Showing $\lbrace (x,y) \in \mathbb{R}^2:xy=1 \rbrace$ is Closed

Let $K=\lbrace (x,y) \in \mathbb{R}^2:xy=1 \rbrace \subseteq \mathbb{R}^2$. Show that $K$ is closed. I am following Munkres' topology book, and this is a step towards finishing problem 3 on p. ...
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1answer
25 views

prove finite intersection property for compact sets using sequential compactness

Prove finite intersection property for compact sets in metric spaces using sequential compactness with a direct proof . One approach is to prove sequential compactness and covering compactness are ...
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0answers
24 views

converging subsequences of two metrics

if $d$ and $d'$ are two metrics on a space $X$, is it true that they induce the same topology if and only if they have the same converging sequences ?
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1answer
39 views

Why is there a subsequence of $(x_n)$ that converges to some point $y$ in $\mathbb R^p$?

A subset $A\subseteq\mathbb R^p$ is compact iff for every sequence $(x_n)$ in $A$ there is a subsequence $(x_{n_k})$ which converges to a point of $A$. I understand the whole proof of the above ...
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0answers
15 views

Definition of normal sets and compactness

I am struggling a little bit with this notion. In Conway's Functions of One Complex Variable, he offers the definition: A set $\mathscr F \subset C(G,\Omega)$ is "normal" if each sequence in ...
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3answers
61 views

Open sets and compact spaces

I am reading through Rudin's Principles of Mathematical Analysis and had a few related questions. First, Rudin defines an open set, $E$, as a set such that every point is an interior point. A point ...
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1answer
35 views

A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
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1answer
20 views

Continuity of multivariable functions

I have a question regarding norms on $\Bbb R^{n}$ and proving the continuity of multivariable functions. Specifically, suppose we have $f: \Bbb R^{2} \to \Bbb R$, for example. To prove $f$ is ...
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2answers
51 views

Show that $A$ is open in $\mathbb R$

I got this question in a test earlier today. I know it is a very small question, since it only counted 2 marks, but for some reason I simply could not get it?? Let $f:\mathbb R \to \mathbb R$ be ...