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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Metrizable and First Countable Spaces

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. While studying Topological Spaces, I came across metrizable spaces. If I understand this ...
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21 views

Homotopy equivalence between O-O and $\theta$

Show that the dumbbell O-O (where there's no space between the "O" and "-") and the letter $\theta$ are homotopy equivalent, using the definition. So, let $X$ be the set of points in the dumbbell, ...
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55 views

Armstrong's “Basic Topology” proof of Tietze Extension Theorem is wrong - what's the best fix?

In Armstrong's "Basic Topology" he proves the Tietze Extension Theorem by first defining $d(x,A)$ (the distance from a point $x$ to a set $A$) as the infimum of a numbers $d(x,a)$ where $a\in A$. He ...
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41 views

Side identification on a hexagon

Apparently giving a hexagon side identification aabbcc results in a sphere. I'm struggling to see this, can someone explain? perhaps with a diagram? It seems to be all the vertices are identified, but ...
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35 views

Every order topology is regular (proof check)

My proof: Let $X$ be an space with the order topology, $x \in X$ and $F$ a closed set that does not contain $x$. Then, the set $X-F$ is an open set that contains $x$, hence there is an open set ...
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18 views

Minimal conditions for compactness of PDFs

I need to find some set of (minimal) conditions to put on a family of probability density functions with bounded support so that the family becomes compact. (I want to use Sion's theorem, which ...
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15 views

Dimension of a subgroup of a solenoid with measure zero

Let $G$ be a connected compact finite-dimensional abelian group (also called a solenoid). If $H$ is a subgroup of $G$ with Haar measure $0$, can we say something about the connectedness or the ...
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25 views

Non-trivial convergent sequences and continuous funtctions on toplogical spaces

Let $X$ and $Y$ be topoligical spaces. We say that $S\subset\ X$ is a non-trivial convergent sequence of $X$ if the following three conditions are fulfilled: i) ...
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29 views

How to make sure any two points with small enough distance are inside a common open set

Let $K$ be a compact subset of a metric space, and I cover $K$ with finite open sets. How can I select $\delta>0$, such that for any $x,y\in K$, with $d(x,y) <\delta$, $x,y$ are inside an open ...
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52 views

Homeomorphisms and Preorders on Topological Spaces

This is a follow up question to this question. It's not difficult,but I'm curious. Let X, Y be homeomorphic topological spaces and let ~ be a preorder on X. Then in general, wouldn't the relation be ...
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13 views

Question regarding uniform spaces and equicontinuity number 2

Following the already answered question: Question regarding uniform spaces and equicontinuity in the context of proposition 27. How do we know that indeed every element in the p-closure of G is a ...
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21 views

Dual group endowed with the compact-open topology

I wanted to ask a question. Let $G^*$ be the dual group of an abelian topological group $G$ ($G^*$ is defined to be the group of all continuous homomorphisms from $G$ to the circle group $T$). I ...
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23 views

Isomorphism of finite dimensional topological vector space with $(\mathbf{R}^k,\mathcal{R})$

Let $(T,\mathcal{T})$ be a topological vector space over $\mathbf{R}$ with finite positive dimension. Is it true that there exists an isomorphism between $(T,\mathcal{T})$ and ...
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36 views

homomorphism inducing Galois cover

We are given a homomorphism $\rho: \pi_1(\Sigma_g - \{p_1,...,p_k\}) \rightarrow G$ , where $\Sigma_g$ is genus g Riemann surface and $p_i$'s are points on it, G is a finite group. Then it is claimed ...
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19 views

Lifting property of a covering map, product topology version

Suppose I have the following theorem (1): If $C,X$ are spaces, $p:C\to X$ is a covering map, $Y$ is a "nice" topological space (I think simply connected and locally path-connected is sufficient), ...
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38 views

A set $U$ is open iff it is union of open balls

Let $(X,d)$ be a metric space. Consider the collection $\mathcal{T} = \{ U \subset X: \forall u \in U, \exists r>0 \; \; , B_r(u) \subset U \}$. We showed that $(X, \mathcal{T} )$ is a topological ...
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32 views

What kind of topology is this (if it involves non-set operations)

Suppose we have $\cup, \cap$ defined as binary operations on "subsets" of some set $S$, where "subsets" can be something exotic like substrings of a string. Suppose also that they satisfy analogous ...
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19 views

Clarification needed for an example about Hausdorff space

This is from Kolmogorov & Fomin Introductory Real Analysis[p.86]. Example 4. Consider the closed unit interval $[0,1]$, where neighborhoods of any point $x\neq 0$ are defined in the usual way ...
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28 views

The “Wiggle Room” intuition for cofibrations

Often enough - for instance in the answer to this question - I have encountered the idea that an inclusion $i:A\subset X$ is a cofibration if $A$ has enough "wiggle room" in $X$. Although I have a ...
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31 views

Monodromy Theorem and homeomorpihsm

If (A,f) is a smooth unlimited covering surface of X, f maps A onto X, and X is simply connected, then the Monodromy theorem implies f is a homeomorphism? I can't see this totally. Do you have any ...
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49 views

Two norms $||.||_1$ and $||.||_2$ on a vector space $V$ are equivalent.

Two norms $||.||_1$ and $||.||_2$ on a vector space $V$ are equivalent iff there exist positive constants $C_1,C_2$ such that $$C_1||.||_1 \leq ||.||_2 \leq C_2||.||_1$$ for all $x \in V$. I have ...
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22 views

Homeomorphism of a difference

Suppose we have $A\approx B(0,2)$ and $B\approx B(0,1)$, with $B(0,r)=\{x\in\mathbb{R}^n\mid \lVert x\rVert<r\}$. If $A\backslash B\neq \emptyset$, is $A\backslash B$ homeomorphic to ...
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33 views

Why is $\mathcal{C}(X,G)$ a top. group via the co. topology?

Let $X$ be an arbitrary topological space and $G$ a topological group. Let $\mathcal{C}(X,G)$ be the group of continuous maps from $X$ to $G$, endowed with the compact-open topology as a topological ...
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55 views

Classify Maps up to Homotopy from Special Cylinder to $S^4$

Let $n\in\{0,1,2,3,4,5\}$ be given. Let $F \subseteq \{1,2,3,4,5\}$ be given such that $|F| = n$. Define an $F$-flip map on $S^4\to S^4$ by sending $x_j \stackrel{F\mbox{-flip}}{\mapsto} ...
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44 views

$(\Bbb N, d)$ and $(\Bbb N, \delta)$ are homeomorphic.

Let $\Bbb N \subset \Bbb R$ be given the induced euclidean metric $d$ and we consider $\Bbb N$ with the discrete metric $\delta$. To show: $(\Bbb N, d)$ and $(\Bbb N, \delta)$ are homeomorphic. I ...
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37 views

Classifying topological spaces

I'm trying to produce a diagram which illustrates the relationships between different topological spaces; from the 'simplest' to the increasingly more structured. However, I'm not certain about some ...
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83 views

Identify subsets of $\mathbb{N}$ with their characteristic functions

If $G\subseteq2^{\mathbb{N}}$ is comeager then exist is a partition $\mathbb{N}=A_0\cup A_1$, $A_0\cap A_1=\emptyset$, and sets $B_i \subseteq A_i$ for $i \in \{0,1\}$, such that for $A \subseteq ...
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52 views

Borel sigma algebra, generated by borel functions, representation of a point

Please let me ask a question about $\sigma$-algebra. Let $E$ be a Hausdorff space and $\mathcal{B}(E)$ denotes its Borel $\sigma$-algebra. Let $(u_{n})_{n \in \mathbb{N}}$ be a family of ...
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34 views

Limit points of a subset of a topological space.

Determine the set of limit points of $A=\{1/n+1/m : n,m \in \mathbb Z^+\}$ in the standard topology on $\mathbb R$. I think that the limit points of $A$ is $A'=\{0\}$. Am I correct? How would I prove ...
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25 views

About the weak* closedness of the kernel of a continuous linear functional

I will appreciate any help on the following: Let $X$ be any Banach space (complex or real) and $\varphi\in X^{**}$ with the property that $\ker(\varphi)$ is weak$*$ closed. I want to prove that ...
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43 views

Fundamental group of quotient disk

Consider the disk $D^{2}$ in $\mathbb{R}^{2}$. By taking out two disjoint, smaller disks within $D^{2}$, we obtain a disk of genus 2. Now consider identifying the boundaries of the two deleted circles ...
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24 views

Double mapping cylinder- a point set question

We have the following set up: $X_0 \subset X_{\pm} \subset X$. Also interiors of $X_\pm$ cover $X$. Now let $Z$ be the double mapping cylinder of the maps $X_- \leftarrow X_0 \rightarrow X_+$. Define ...
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40 views

Show a closed, convex, absorbing set in a Topological space nonmeger in its self contains a neighborhood of $0$.

Been sitting on this one for a few days and would really appreciate some help. I have included a definition and theorem that seemed useful. If anyone would be willing to critique or confirm my proof ...
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27 views

alternating proof for completion theorem

Instead of using equivalent class as is used in pugh's real mathematical analysis. If I have proved every metric space $S$ has a isometric copy $S_0$ in $C^0(M,R)$. And since $C^0(M,R)$ is complete, ...
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23 views

How to define a compactly generated space?

I engaged two definitions for a compactly generated space: http://en.wikipedia.org/wiki/Compactly_generated_space 1) In topology, a compactly generated space (or k-space) is a topological space ...
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30 views

Borel Measures: Riesz-Markov-Kakutani

Problem Given a locally compact Hausdorff space. Consider a positive functional: $$I:\mathcal{C}_0(\Omega)\to\mathbb{C}:\quad f\geq0\implies I f\geq0$$ Then it has a representation via a regular ...
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24 views

Continous dependence of the minimium of a continuous function over a compact set

Suppose we are working on $\mathbb{C}^n$ and $h = (c_1,\dots,c_k)$ is a unit vector $$ |c_1|^2 + \dots + |c_k|^2 = 1 $$ Now consider the function for $t\geq0$ $$ f(t,h) = \left(\sum_{i=1}^k ...
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80 views

Proof of a lemma is unclear to me (Theorem of inverse functions)

Lemma: Let $B(a,r)$ be a ball in Banach space $X$ and $\phi$ be a contraction ($d(\phi(x),\phi(y)\leq qd(x,y),0<q<1$) from $B(a,r)\to X $. Then the function:$$ f(x)=x+\phi(x)$$ is a ...
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33 views

Local bisections of Lie groupoids

Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...
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28 views

A simply connected region $D$ that contains the boundary of $S$ contains $S$

If $D\subseteq X$ is a simply connected subspace of the topological space $X$ and [add assumptions here] and $S\subseteq X$, $\partial(S)\subseteq D$, then $S\subseteq D$. It doesn't seem to be ...
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13 views

which space hosts space of probability distributions with weak topology?

Given a separable metric space $A$, let $P(A)$ be the space of probabilities defined on $A$ along with its Borel sigma field. One can define Prohorov metric on $P(A)$, which induces the weak topology. ...
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39 views

Set $P$ is dense in $(0,1)$

Let $p >1$ be a prime number. Prove that $$P : = \{ \sum_{j=1}^{k} \frac{a_j}{p^j} \, ; \, a_i \in \{0, \ldots, p-1\}, k \in \mathbb{N} \}$$ is dense in $(0,1)$. Attempted solution: Note ...
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36 views

Arbitrary union of connected subsets Follands

$\textbf{Question:}$ If $\{E_{\alpha} \}_{\alpha \in A}$ is a collection of connected subsets of $X$ such that $\bigcap_{\alpha \in A} E_{\alpha} \neq \emptyset$, then $ \bigcup_{\alpha \in A} ...
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53 views

Weak * topology on a finite-dimensional simplex

I'm trying to endow a set of probability measures $\triangle\left(X\right) $ with the weak * topology, where $X=\left\{ x_{1},\, x_{2},\,...,\, x_{N}\right\} \subseteq\mathbb{R}$ is a finite set of ...
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26 views

finding homology of disjoint closed balls

If $U \subset \mathbb{R}^m$ and $B$ is a closed ball contained in $\mathring{U}$ (interior of $U$) with $B_1,B_2,\ldots,B_n$ are disjoint closed balls contained in $\mathring{B}$ (interior of $B$). ...
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51 views

Mapping $X$ to $\text{sk}_2(X)$ while fixing $\text{sk}_1(X)$

UPDATE: The original question can be simplified to this: Given a finite simplicial complex $X$, can I find a continuous $f : X \rightarrow \text{skel}^2(X)$ that fixes $\text{skel}^1(X)$? Basically, ...
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21 views

Greatest Lower Bound and Distance

The following question is an exercise from Fred H. Croom's book "Principles of Topology." Let $x$ be a real number and $A$ a subset of $\mathbb{R}$. (a) Prove that if $d(x,A)>0$, ...
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36 views

Metrizibality of an uncountable product of the real line

Below is a question from Topology, James Munkres. Following that is my attempt at a solution, which I am not sure is correct and would appreciate if somebody could point out what (if anything) is ...
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37 views

Contraction with different metrics

Is it possible for a function that isnt a contraction with the euclidean metric, to be a contraction using a different metric? (I can only think of the opposite) Are there any examples of such ...
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35 views

Direct limits of locally convex spaces and embeddings

I was thinking about whether this positive result would hold in the category of locally convex spaces also... Here is what I got so far: The direct limit of a locally convex system consists of the ...