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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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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34 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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34 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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54 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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37 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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46 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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28 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 , ...
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1answer
30 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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44 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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39 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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29 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 ...
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72 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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64 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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431 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 ...
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1answer
80 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 ...
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61 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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71 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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102 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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66 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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27 views

If $\overline{A}\cap \overline{B}=\emptyset$, then $\partial(A\cup B)=\partial A\cup \partial B$ [duplicate]

If $A$ and $B$ are two subsets of a topological space $X$ such that $\overline{A}\cap \overline{B}=\emptyset$, then $$\partial(A\cup B)=\partial A\cup \partial B$$ So I need to prove that: ...
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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 ...
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1answer
80 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
60 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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99 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 ...
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1answer
33 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
30 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
18 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
19 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 ...
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1answer
79 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 ...
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90 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 $$ ...
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1answer
37 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
121 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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96 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
32 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.
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2answers
41 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 ...
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730 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.
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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
65 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 ...
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1answer
41 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?
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1answer
68 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 ...
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3answers
63 views

homeomorphism between topological space and product space

Is there any connected topological space $X$ such that $X$ is homeomorphic to $X\times X$ ?
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29 views

Embeddings and Characteristic Property of Subspaces

Any embedding obeys the char. prop. of subspaces: "If $X_0\cong A\leq X$ then $X_0\hookrightarrow X$" Does the converse hold as well: "If $X_0\hookrightarrow X$ then $X_0\cong \mathrm{im}\leq X$" ...
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206 views

What does Structure-Preserving mean?

A very basic definition in category theory is the definition of morphism between objects. If the category is a construct, i.e., a category $\mathcal C$ equipped with a faithful functor $U\colon ...
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Characteristic Property = Universal Property?

Problem They seem to be the same -almost! But are they really or is it just unlucky accident that they look so similar however describe totally different notions? Example I was trying to set the ...
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1answer
51 views

Every Bounded set contained in a Compact set

In a general metric space, is every bounded set contained in a compact set?
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1answer
89 views

Why topological embedding continuous?

Problem Why do we require a topological embedding to be continuous? Compared to other categories we want a space to be isomorphic to some subspace. Translating this to topological spaces that is ...
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Theorem 23.5 from topology by James Munkres

I'm reading the following theorem: I don't understand what would go wrong if you would leave out the part that I marked green. I would think you could safely leave it out and replace the second to ...
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A more general definition of branched covering.

If $f:X\longrightarrow Y$ is a holomorphic map between two compact Riemann surfaces, then $f$ is called also a branched covering map. This because the branched points of $f$ form a finite set ...
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57 views

Closure of a set is the set of limit points.

Let $A \subseteq \mathbb{R}^n$. Let $S = \{ x \in \mathbb{R}^n : \exists (x_n) \subseteq A \; \; \; s.t \; \; x_n \to x \} $ $$ \text{Claim}: \; \overline{A} = S $$ Attempt $ \overline{A} \subseteq ...
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How can I understand the three-dimensional space forms?

Here is what I know: A space form is defined as a manifold admitting a Riemannian manifold of constant sectional curvature A classical result of Cartan states that a manifold is a space form if and ...