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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5
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0answers
27 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 ...
6
votes
1answer
383 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 ...
4
votes
0answers
16 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?
4
votes
1answer
77 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 ...
0
votes
0answers
15 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 ...
2
votes
0answers
19 views

Does the product functor preserve quotient maps?

In Hatcher's Albebraic 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 ...
0
votes
0answers
29 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 ...
2
votes
0answers
25 views

The fundamental group of $S^n$

I want to prove that $\pi_1(S^n,x_0)$ is trivial if $2\leq n,$ BUT using universal covering. So let $p:\tilde S^n \rightarrow \tilde S^n$ the universal covering. Define $f:D^n\rightarrow S^n$ such ...
0
votes
1answer
32 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 ...
0
votes
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 ...
1
vote
2answers
25 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?
22
votes
4answers
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 ...
1
vote
2answers
22 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 ...
0
votes
1answer
14 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 ...
3
votes
0answers
16 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
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: $(\Rightarrow)$ ...
0
votes
1answer
25 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 ...
-3
votes
1answer
46 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 ...
-2
votes
1answer
52 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 ...
1
vote
1answer
63 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. ...
0
votes
4answers
32 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 ...
2
votes
1answer
319 views

two problem on homeomorphism and product of topological spaces

Let $\{(X_i, T_i) : i = 1,2,\ldots, n\}$ be a collection of topological spaces and let $\sigma$ be a permutation of the symbols $1, 2,\ldots, n$. For $i=1,2,\ldots, n$ let $Y_i = X_{ \sigma (i)}$ and ...
5
votes
2answers
388 views

A zero-dimensional Hausdorff space is totally disconnected

The full question: A space is zero-dimensional if the clopen subsets form a basis for the topology. Show that a zero-dimensional Hausdorff space is totally disconnected. Recall a space is totally ...
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
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
votes
1answer
34 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 ...
2
votes
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 ...
1
vote
1answer
34 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?
2
votes
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 ...
2
votes
2answers
123 views

$S^2$ with countably many points removed is path-connected

Prove that after removing a countably infinite number of points from $S^2$, it remains path-connected. This was a question that arose in the algebraic topology course I have this term. I thought ...
0
votes
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 ...
2
votes
3answers
48 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. ...
6
votes
1answer
89 views

Characterization of open maps in terms of nets

Here I asked about characterization of closed maps in terms of nets/sequences. I find this view illuminating, so I wanted to ask about open maps. A map $f: X \to Y$ is open if for each $x \in X$ and ...
1
vote
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 ...
3
votes
4answers
91 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. ...
1
vote
1answer
95 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 ...
2
votes
3answers
65 views

Prove that the set $\{(x,y) \in\mathbb R^2\mid 2<x^2+y^2<4\}$ is an open set

I've been trying to prove this in the following way: Name the area between the two circles $S$ Suppose $(x,y) \in S$. Let $\delta = \min \{\sqrt{4-x^2-y^2},\sqrt{x^2+y^2-2}\}$. Then, I'm trying to ...
0
votes
0answers
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 ...
2
votes
1answer
22 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 ...
5
votes
1answer
355 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
1
vote
1answer
118 views

Trying to complete a proof for my real analysis course (total boundedness)

I'm trying to prove that any totally bounded subset $S$ of a metric space $X$ contains finitely many points such that the union of the open epsilon balls centered at these points includes the set $S$. ...
-1
votes
1answer
38 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 ...
7
votes
1answer
47 views

Irreducible projective cubic, exists continuous surjection?

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
0
votes
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 ?
1
vote
1answer
39 views

An exhaustive continuous map is a covering map.

$p_1:\tilde X_1 \rightarrow X \, ; \, p_2:\tilde X_2 \rightarrow X$ two coverings maps, where $X$ connected and locally path-connected, and suppose that $f:\tilde X_1 \rightarrow \tilde X_2$ is an ...
0
votes
0answers
14 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 ...
0
votes
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 ...
2
votes
1answer
36 views

Show that if $f$ is a proper,surjective map which is locally injective then $f$ must be a covering map

Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map. It is well ...
1
vote
1answer
78 views

Convexity implies the equivalence of order and subspace topologies

From Munkres p.90-91 Definition Given an ordered set $X$, a subset $Y$ of $X$ is convex in $X$ if for each pair of points $a<b$ in $Y$, the entire interval $(a,b)$ of points of $X$ lies in $Y$. ...
1
vote
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 ...