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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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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46 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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27 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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53 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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43 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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48 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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41 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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20 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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24 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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26 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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22 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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27 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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23 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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76 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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32 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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24 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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11 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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32 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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25 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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50 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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20 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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33 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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35 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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33 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 ...
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26 views

A question about the dimension theory of Hilbert space.

Let H be a real, separable and infinite dimensional Hilbert space. If S is any subset of H that disconnects H, is S necessarily infinite dimensional? I cannot find any discussion of this in the ...
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35 views

Irrational winding of a torus

Let $f: \mathbb{R} \mapsto T=\mathbb{R}^2/\mathbb{Z}^2$ such that $x \mapsto (x,\alpha x) \, \, \mod \mathbb{Z}^2$ and $\alpha$ irrational. I want to prove that $f$ isn't an embedding. In order to do ...
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22 views

Tikhonov theorem and $L^1$ completeness

The idea is to prove completeness of $L^1$ using Tikhonov theorem. The proof will be for narrower class of functions though. There is a Tikhonov theorem that states that for any set of compact ...
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28 views

Open cover of manifold with boundary

If $\mathcal{O}=(U_{\alpha})_{\alpha\in A}$ is an open cover for a smooth manifold $M$, then each $U_{\alpha}$ is a smooth manifold. I want to extend this fact to manifolds without boundary. So my ...
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99 views

Connectedness of the Complement of the Countable Union of Closed Jordan Regions

Let $\{V_j\}$ be a countable collection of pairwise disjoint closed Jordan regions in $\mathbb{R}^2$. That is closed sets whose boundary is a Jordan curve. Let $$U = \mathbb{R}^2 \setminus \bigcup ...
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42 views

Perfect preimages of compact spaces are compact

Below is a question concerning compact spaces from James Munkres' Topology. Following that is my attempt at a solution, which I am not sure is correct and would appreciate if somebody could point out ...
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31 views

Isomorphisms with invariant linearly independent dense subset.

If $T$ is an isomorphism acting on a separable Banach space $X$, can we find a countable, dense, linearly independent set $D\subset X$ such that $T(D)=D$? If $X$ is finite dimensional, then the answer ...
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42 views

about Heine-Borel Theorem in a function space

In Pugh's real mathematical analysis. About the Heine-Borel Theorem in a function space, it states that a subset $\epsilon$ $\in C^0$ is compact if and only if it is closed, bounded, and ...
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33 views

Can someone intuitively describe the fiber bundle and product spaces of SO(3)?

I have zero understanding of differential geometry or topology so the material found online are useless for me. So in light of that can someone use very general terms or analogy to comment about the ...
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27 views

homotopy between continuous functions to an absolute retract

I have the following statement to prove as one of the "fundamental" questions our topology professor wants us to know for his final: Let $X$ be a topological space, and let $A$ be an absolute ...
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Parametrizing regions of complex plane

Let $\Omega=\mathbb{C}\setminus \lbrace t e^{it} \ \vert t \in \mathbb{R}_{\geq0} \rbrace$ I need to write $\Omega= \coprod_{i=0}^{\infty} R_i$ where each $R_i$ is the region bounded by from $t=2k ...
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20 views

Two non-homeomorphic spaces with continuos bijective functions in both directions

I was asked the following question: if two topological spaces $X, Y$ are such that there exist a function $f:X\rightarrow Y$ continuos and bijective and a function $g:Y\rightarrow X$ continuous and ...
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30 views

Density character of a metric subspaces

Is it true that if $M$ is a metric space and $N$ is a metric subspace of $M$ (I mean, $N\subseteq M$ and the metric defined on $N$ is the same metric on $M$ restricted to $N$) then the density ...
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29 views

Topology over $C^0(\mathbb{R})$

Let $C^0(\mathbb{R})$ be the set of continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$, For any continuous function $h > 0$ consider $B_f(h) = \{ g \in C^0(\mathbb{R}) : |f(x) - g(x) ...
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68 views

Equivalence of sigma algebras on the set of probability measures.

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and ...
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33 views

how to conclude a subset of $M_n(\mathbb{C})$ is compact from spectral radius?

could any one tell me which of the following is/are compact subset? $S=\{A\in M_n(\mathbb{C}): \rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): A=A^*,\rho (A)\le 1\}$ $S=\{A\in M_n(\mathbb{C}): ...
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23 views

Entropy of isometric extension

A similar question to mine was asked before at the address below but it was not answered there so I am asking it again. Also there is a more specific case I am interested in. Topological entropy of ...
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35 views

Topological spaces from compact Hausdorff zero dimensional spaces

I saw a construction of general topological spaces using compact Hausdorff zero dimensional topological spaces, but I have no clue now of the construction or reference to this. I would be thankful if ...
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30 views

Test functions are dense in $L^p$?

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a ...