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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59 views

Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
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1answer
148 views

Definition of compact subspace of a metric space

Given a metric space $X$, we say $X$ is compact iff there is an open cover $\{U_\alpha\}_{\alpha \in A}$ of $X$ with a finite subcover, that is to say there is a finite subset $A'$ och $A$ such that ...
3
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1answer
94 views

How to prove the uniqueness of a continuous extension of a densely defined function? [duplicate]

Let $X$ be a topological space and let $Y$ be a Hausdorff space. Let $D$ be dense in $X$. Prove that continuous functions $f, g : X \to Y$ which are equal in $D$ are equal in all $X$. I'm a little ...
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4answers
87 views

topology about isolated and limit point

As a first class in topology, it is hard to prove. Can you help me? $S\subset X^{metric}$. Let $S_1$ be the set of limit points of S. Let $S_2$ be the set of isolated points of S. Show that $\bar S = ...
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1answer
72 views

Show that the closure of S in Y coincides with $\bar S \cap Y$

As a novice, it is hard to prove the following problem. Let Y be a subspace of a metric space X and let S be a subset of Y. Show that the closure of S in Y coincides with $\bar S \cap Y$, where $\bar ...
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0answers
76 views

Connected subsets of the closed unit disc

I have been trying to solve the following problem : Does there exist two disjoint connected subsets of the closed unit disc in $\mathbb R^2$ such that one contains the points $(1,0)$ and $(-1,0)$ and ...
3
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1answer
78 views

Questions on “simple-connectedness-like” property

I wanted to know if there's any notion which is very similar to the simple connectedness, but defined "purely" in terms of points and sets. Here's my attempt to do it. Let $X$ be a topological space. ...
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2answers
173 views

Compactness of Topological Spaces

The only one that I have been able to get at is that (c) is not compact since it is not closed/bounded by Heine-Borel Theorem. Any thoughts on how to approach the others? I understand that, in order ...
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1answer
71 views

If $X$ is homeomorphic to $Y$, does every map $X \to X$ factor through a map $Y \to Y$?

Let $X$ and $Y$ be topological spaces, and $h:X \to Y$ a homeomorphism. For every continuous map $f:X \to X$, is there a continuous map $g:Y \to Y$ such that $f=h^{-1} \circ g \circ h$? This came up ...
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1answer
951 views

Stereographic projection is a homeomorphism $S^n \setminus \{p\} \to \mathbb{R}^n$

Let $S^n$ be the $n$-sphere, $N=(0,0,...,0,1)$ and be the north pole of $S^n$. I am trying to show that stereographic projection gives a homeomorphism $\sigma: S^n \setminus \{N\} \to \mathbb{R}^n$. ...
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1answer
52 views

Directly Proving a Set is Closed

Is there a good way to prove that some random closed set is closed using the "contains it's limit points" definition? For instance how could you directly prove the first quadrant with the positive x ...
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1answer
51 views

Extension of a metric from closed subspace of a metrizable space to the whole space

How to prove: Let M be a closed subspace of a metrizable space X. Then, any metric on the subspace M can be extended to a metric on X.
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2answers
60 views

Distance between a point and a set

The problem I'm trying to solve is Prove that $d(a, B \cup C)$ is the smaller of $d(a,B)$ and $d(a,C)$ for a point $a$ and subsets $B, C$ of a metric space. So I think what I need to show is ...
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1answer
105 views

Boolean Closure and Borel sets

Denote the boolean closure of a family of sets $\mathcal S$ by $\mathcal B(\mathcal F)$, then in a metric space it is well known that $\mathcal B(\mathcal F) = \mathcal B(\mathcal G) = \mathcal ...
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2answers
72 views

A open subset of $\Bbb R$

Given the definitions in Open Subsets of open sets I need to prove that $\{x \in \Bbb R : |x|>2\}$ is open in $(\Bbb R , d_E)$ This seems to be true, however I don't know how to prove it without ...
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1answer
39 views

Quick topology question

I confused myself. It is a seemingly trivial question: If $U,V,B$ are sets in a topological space $X$ and $U \subset B$ is open in $B$ and $U = U \cap V$ is it true that $U \cap V$ is open in $B \cap ...
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2answers
74 views

Can $\overline{Y}$ have non-empty interior if $Y$ has empty interior

(below, $\overline{Y}$ denotes the closure of $Y$) Given a metric space $X$ let us define a subset $Y$ to be nowhere-dense if and only if $\overline{Y}$ has empty interior. It is obvious that if ...
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0answers
52 views

Open Book Decompositions of 3-manifold and Associated Heegard Splittings

In page 13 of the paper: http://arxiv.org/pdf/math/0510639v1.pdf It is stated that "An open book decomposition (S,h, K) , gives rise to a special Heegard decomposition of M ". Here, S is a surface , ...
2
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1answer
48 views

Continuous map of a compact set

Claim: If $f:X \to Y$ is continuous, where $X$ is compact, and $Y$ is Hausdorff, then $f$ is a closed map. Proof: Take $A \subset X$ to be closed in $X$. Now as $X$ is compact and by choice of $A$ we ...
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1answer
101 views

Munkres: Compact subsets of Hausdorff Space

Claim:If $A,B$ are compact disjoint subsets of the Hausdorff space $X$, then there exists disjoint open sets $U,V$ containing $A,B$ resp. Would I be on the right track in saying that since $A,B$ are ...
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2answers
92 views

Munkres: Connected Sets in the Real Line

On page 154 Munkres proves the Intermediate Value Theorem and there is one part that I am unclear of. He constructs the two set $A=f(X) \cap (-\infty,r)$ and $B=f(X) \cap (r,+ \infty)$ This ...
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1answer
72 views

Order-preserving embeddings

(Follow-up to Existence of a utility function on the reals.) Say we have a totally ordered set $X$ which has a countable, dense subset $C$. I believe we can find an $f:C\to\mathbb R$ which is ...
4
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1answer
84 views

Metric space in Topology class

On the set of integers $\mathbb Z$, show that the function d, defined as follow, is a metric : $$ d(x,y) = \begin{cases} 0 & \text{if } x=y \\ \min\{1/n! \mid n! \text{ divides } |x-y|\} & ...
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2answers
111 views

Continuous images of open sets

In trying to prove that the graph of a continuous map of compact Hausdorff spaces, $f:X\to Y$ is compact, I stumbled on this problem: Let $f:X\to Y$ be a continuous function, $U$ and $V$ open in $X$ ...
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2answers
647 views

How big can a separable Hausdorff space be?

It is just an idea (might be wrong) but, i think that if a Hausdorff space, say $X$, contains too many elements, then a countable subset cannot be dense in it. Does there exist a cardinality that any ...
2
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1answer
126 views

In $\mathbb{R}^n$ how many disjoint open sets can share a boundary

I know that in $\mathbb{R}$ you can have at most $2$ disjoint open sets that share a boundary(I believe my answer to Open Sets Boundary question proves that). My question is is there a way to extend ...
3
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2answers
223 views

closed ball in euclidean space

In general metric spaces the closed ball is not the closure of an open ball. However, I read that in the Euclidean space with usual metric, closed ball is the closure of an open ball. I'm having ...
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1answer
144 views

The Solution to Exercise 4, page 12, of Gamelin's “Introduction to Topology”.

I'm looking at exercise 4 in page 12 of Gamelin's Introduction to Topology. The problem is stated as follows: Suppose that $F$ is a subset of the first category in a metric space $X$ and $E$ is ...
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0answers
243 views

Wedding Vows puzzle

My father came up with a puzzle and dared me to solve it. I could solve it by trial and error, but I rather want to solve it mathematically. It is the so called "Wedding Vows puzzle" where you have to ...
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3answers
68 views

Show that $\{y:I\longrightarrow Y \mid y(j)=y(i) \;\; \forall j\in I\}$ is closed.

Let $I$ be a non-empty set and $Y$ be a Hausdorff space. Fix $i\in I$ and define $$D:=\{y:I\longrightarrow Y \mid y(j)=y(i) \;\; \forall j\in I\}.$$ Show that $D$ is a closed subset of ...
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4answers
53 views

Are the spaces below homeomorphic?

Is it true that $$X=\{(x, y)\in\mathbb R^2: 0<x^2+y^2\leq 1\}$$ is homeomorphic to $$Y=\{(x, y, z)\in\mathbb R^3: x^2+y^2=1, 0<z\leq 1\}?$$ I was supposed to show it but I can't see ...
2
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1answer
149 views

Quotient Map vs Embedding (Topology)

Problem 1: Can any quotient $\tilde{X}$ of $X$ be embedded in $X$? Moreover, does any (surjective) quotient map $\pi:X\to\tilde{X}$ left split with an (injective) embedding $\iota:\tilde{X}\to X$? ...
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1answer
214 views

corollary to baire category theorem

I'm studying topology with gamelin and greene's text and I came across a corollary to the baire category theorem which states that "Let (En) be a sequence of nowhere dense subsets of a complete ...
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1answer
135 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
0
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1answer
77 views

Surface with border is homotopy equivalent to bouquet of circles

Why is any compact surface with non-trivial boundary homotopy equivalent to bouquet of circles? It was mentined in "Course homotopy topology" by Fomenko, Fuchs while calculating homotopy groups of ...
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1answer
45 views

compact set has a countable base

Let $K \subseteq \mathbb{R}^n$ be a compact set. Then, there exists a countable set $S \subseteq K $ such that $\overline{S} = K$ My try: Notice for any $n$, the collection $U_n = \{ B( x, ...
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1answer
24 views

Question about compact sets and coverings

Let $K \subseteq \mathbb{R}^n$ be sequentially compact. Then for every $\epsilon > 0$ there exists $x_1,...,x_m \in K $ such that $K \subseteq \bigcup_{i=1}^n B(x_i, \epsilon ) $. Proof Suppose ...
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1answer
23 views

Covering of a universal space and covering of a subset

It seems to me that the definition of covering depends on the set we are referring to. For example, if $X$ is the universal topological space, then a collection $\mathcal A$ of subsets of $X$ is said ...
4
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1answer
101 views

Hard to find counterexample for $\partial (\partial A) = \partial A$

In an exercise I've proven that $\partial(\partial A) \subset \partial A$, for any $A\subset X$, where $X$ is a topological space and $\partial$ in this case stands for the boundary. Apparently, in ...
2
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3answers
357 views

show the supremum of the distance function of a compact metric space is finite

Let $X$ be a compact topological space and $(Y,d)$ be a metric space. Show that for every pair of continuous functions $f\colon X\to Y$ and $g\colon X\to Y$, the extended real number $$ ...
1
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1answer
90 views

Show the supremum of a set is less than infinity

Let X be a compact topological space and $<Y,d>$ be a metric space. Show that for every pair of continuous functions $f:X\to Y$ and $g:X \to Y$, the extended real number $$B=\sup\{d(f(x_1), ...
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2answers
317 views

Proof: Categorical Product = Topological Product

Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces $X_i$ and any topological ...
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2answers
231 views

Discontinuity of the characteristic function

Let $A \subseteq \mathbb{R}^n$. Let $f(x) = \chi_A $ be the characteristic function, and put $D = \{ x : f(x) \; \; \text{is discontinuous} \} $. Then $\partial A = D $. MY try: Let $y \in D $. ...
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1answer
37 views

Completely regular vs regular

I know that regularity doesn't imply completely regular; however, does completely regular imply regularity? I assume it does but I'm not sure.
3
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2answers
120 views

Graph of a continuous function is closed

Let $f: \mathbb{R} \to \mathbb{R}$ be continuous. Then $G = \{ (x, f(x) ) : x \in \mathbb{R} \} $ is a closed set. My try: Suppose $(z_n) = (x_n, f(x_n) ) $ is sequence in $G$ with limit $(x,y)$. We ...
7
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3answers
787 views

Why isn't $(0,1]$ compact?

It is said that $$\bigcup_{n\geq 1}\left(\frac 1n, 1+\frac1n\right)$$ is not compact. Why? Is it because it is not closed? Or am I missing something more? Many thanks.
2
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0answers
95 views

Zer0-dimensional, countable, 1st countable T1 space is metrizable?

Show that every countable, first countable, zero-dimensional T1 space $X$ is metrizable. I know that T1 space means that all its singletons are closed. Also, zero-dimensional means that $X$ has a ...
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1answer
73 views

Given $X×Y$ hausdorff. Show that $X$ hausdorff.

Given $X×Y$ hausdorff. Show that $X$ hausdorff. Assume $x_1≠x_2$ in $X$. Then $(x_1,y_0)≠(x_2,y_0)$ for some $y_0∈Y$. Then there exists disjoint open neighborhoods in $X×Y$. As those neigborhoods ...
0
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1answer
64 views

Is a Covering Space of a Topological Space always Hausdorff?

Is a Covering Space of a Topological Space always Hausdorff? I can separate two different points from the same fiber, but what about two arbitrary points?
2
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1answer
154 views

Contractibility of the space of sections of a fiber bundle

Let $\pi: E \to M$ a fiber bundle and $\Gamma(M,E)$ the space of smooth sections of the bundle with topology induced by the Whitney topology on $C^{\infty}(M,E)$. Assume that each fiber is ...