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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92 views

Hausdorffness of quotient space

Let $G$ be a compact topological group, and $X$ be a Hausdorff space. We assume that $G$ acts on $X$. Is the quotient space $X/G$ with the quotient topology a Hausdorff space? It seems that the ...
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40 views

Discrete Analogue of the Poincaré Conjecture and Simple Connectedness

I apologize if this question is badly worded or obvious, but I have no formal topology background. I have put some effort into trying to find something, but nothing turned up, perhaps due to my lack ...
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74 views

Path Homotopy in a Topological Annulus

Let $C_1$ and $C_2$ be simple, closed curves in $\mathbb{R}^2$ such that $C_1$ lies in the region bounded by $C_2$, and the origin $O$ lies in the region bounded by $C_1$. Define an annulus $A$ as the ...
2
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82 views

How many Y shapes can you fit on the plane?

You can only fit at most countably many disjoint open discs on the plane: for any collection of disjoint open discs, it is possible to pick a single rational coordinate contained in each disc, and ...
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717 views

Introductory Topology True/False check - Topology without tears - Exercises 1.1

I have just started to learn Topology, using specifically the book mentioned in the title. I have placed that information in the title with SEO in mind, if this is not acceptable practice in this ...
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75 views

Is it true that $\operatorname{Int}(A) \cap \operatorname{Int}(B) = \operatorname{Int}(A \cap B)$?

can someone please verify my proof? (a) Is it true that $\operatorname{Int}(A) \cap \operatorname{Int}(B) = \operatorname{Int}(A \cap B)$? (b) Is it true that $\bigcap ...
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36 views

Souslin space and functional

I have a question about Borel $\sigma$-algebra on a Souslin space. Let $E$ be a locally convex topological real vector space which is a Souslin space, that is, the continuous image of separable ...
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161 views

Rudin Theorem 2.7

Theorem 2.7 in Rudin's Real and Complex analysis Theorem Suppose $U$ is open in a locally compact Hausdorff space X, $K \subset U$, and $K$ is compact. Then there is an open set $V$ with compact ...
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87 views

Basic question about lifting maps to covering spaces

Any continuous map $f: X_1 \to X_2$ "lifts" to a map $\tilde f: \tilde X_1 \to \tilde X_2$ (provided that $X_1$ and $X_2$ have universal covers). The space $\tilde X_1$ is certainly ...
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570 views

Proof verification: Munkres exercise 10, section 23

Can someone please verify my proof or offer suggestions for improvement? I'm thoroughly confused by this question, and I'm sure there's a mistake somewhere in my proof. Let $\{X_\alpha\}_{\alpha ...
2
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93 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: ...
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47 views

Ribbon Surfaces and Legendrian Graphs on Contact 3-manifolds.

Let $M=(M, \xi)$ be a contact 3-manifold. I am trying to show that every Legendrian graph L (i.e., a graph embedded in $M$ so that it is everywhere-tangent to the contact planes) admits a ribbon ...
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521 views

Prove that every subset of $\mathbb{R}$ is compact in the finite complement topology.

I need help with my proof in particular. I am aware that there is a similar question elsewhere. Can someone please verify my proof or offer suggestions for improvement? Prove that every subset of ...
2
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32 views

Comparing product topologies

Let $C$ denote the set of complex numbers. Let $T$ be the smallest topology on $C$ such that singletons are closed. Let $T_1$ denote the smallest topology on $C$x$C$ such that all the polynomials in ...
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42 views

Extension of function with values in a Banach space

I want to prove the following Let $E,X$ be Banach spaces, and $Y\subset E$ a closed subspace with codimension $1$. Let $T:Y \to X$ be a continuous linear function. Then there exists a continuous ...
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165 views

At most one connected component of unbounded portion of entire function.

Suppose $f$ is an entire complex analytic function and $R$ a positive real number. Define the set $E:= \{z\in\mathbb{C};|f(z)| < R\}$ to be the set of $z$ whose image is bounded by $R$. I want to ...
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88 views

A Homeomorphism that is not unique even upto Isotopy

I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism ...
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76 views

Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
2
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45 views

Prove that $d_\infty(f, g) = \operatorname{sup}\{|f(x)-g(x)|:x \in [a,b]\}$ defines a metric

Can someone please verify my proof? Let $C[a,b]$ denote the set of all continuous functions from $[a,b]$ to $\mathbb{R}$. Let $d_\infty:C[a,b] \times C[a,b] \longrightarrow [0, \infty)$ be given ...
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61 views

Continuous function - unsure of statement that lacks rigour

I have the following statement in my Topology notes in a section on continuous functions - A polynomial of degree $n$ has at most $n$ roots. Thus $f^{-1}(b)$ is finite. This shows that ...
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61 views

explicit function between transformation matrix and vertex in polyhedron

recently I am stuck in solving a geometric problem. I hope someone could give me some tips, thanks for all in advance!!! Question 1: given a constant polygon $M1$ with 4 vertices: ...
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169 views

Homeomorphism of compact Hausdorff spaces

In the preprint "A REMARK ON CANTOR DERIVATIVE" (http://arxiv.org/pdf/1104.0287v1.pdf), there is the next proof: We show that two countable locally compact Hausdorff spaces $X$ and $Y$ of same ...
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48 views

Connected $G_\delta$ non-singleton, proper subsets in a connected complete metric space with more than one point

This is a question related to my last; I have still not solved it. Maybe this one is easier: Suppose $X$ is a connected complete metric space with more than one point. Must $X$ contain a ...
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63 views

topological spaces with The FDS property?

I have difficulty with bellow Theorem that was asked by one of users: The class of spaces in which all compact subsets are closed is $KC$ space. A space $X$ is said to have the FDS property if each ...
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56 views

Is there a bound for the genus of the generalized petersen graphs?

I've looked online and could only find a bound for specific generalized petersen graphs. Does any bound (lower or upper) depending on $n$ and $k$, where $n$ is the order of a cycle and $k$ is the ...
2
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56 views

isomorphism of algebric torus

I'm trying to prove the following: Let $D_n=(\mathbb{C}^{\times})^n$ (an algebric torus of rank $n$). Assuming $D_k$ is isomorphic to $D_n$ as an algebric group. Prove that $k=n$. So far, I managed ...
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56 views

Turning a cup into a donut: parametrising the continuous deformation

A cup and a donut are topologically equivalent because one can be continuously deformed into another. What if you had a particular cup and a particular donut and wanted to write down (or simulate on a ...
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26 views

Direct sums, tensor products etc. of $G$-vector bundles are again $G$-spaces

Given two $G$-vector bundles $E$ and $F$ over a $G$-space $X$ ($G$ some finite group), I am interested in the vector bundles $E \oplus F$, $E \otimes F$, $\operatorname{Hom}(E,F)$ etc. I am familiar ...
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36 views

an example that property $\delta$ does not imply property $\gamma$

In this article, two properties are mentioned at page 153: property $\gamma$: If $\mathcal U$ is an open $\omega$-cover of $X$, then there is a sequence $G_n \in \mathcal U$, with $\underline {Lim} ...
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25 views

$\omega$-covers and $S_1(\Omega,\Gamma)$ property

I am readund this article and there is some proof in there (top of page 156) which is not clear to me. The definitions are: 1. Property ($\gamma$): If $\mathcal U$ is an $\omega$-cover of $X$, ...
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63 views

Topologies of flag manifolds

I'm currently reading an article discussing flag manifolds and the action of $\mathrm{PSL}(n,\mathbb{C})$ on them. A flag (in my view at least) is a nested sequence $(y^1,\ldots,y^{n-1})$ of subspaces ...
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61 views

Pervin quasi-uniformity

for a subset $ A$ of a set $X$ we set $S_{A} =[(X - A)×X]∪ [X ×A]$. We recall that if $X$ is a topological space, then the Pervin quasi-uniformity of $X$ is generated by the subbase $\{S_{G}: G ...
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34 views

How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a ...
2
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79 views

Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
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50 views

Is there a metric on $\mathbb{R}$ such that $\mathbb{Z}$ is dense in $\mathbb{R}$?

It seems that such a metric doesn't exist,but how to prove?
2
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74 views

How do we understand and visualize the meridians of a 3-sphere?

The equator of a n-sphere is (n-1)-sphere. Then the equator of a 3-sphere is a 2-sphere. Does this mean the meridian of a 3-sphere is a hemisphere (half 2-sphere)? In contrast, Wikipedia describes the ...
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46 views

A dense set and neighbourhood bases induce a topological basis

Looks like I can't get my head around the following proposition. Let $(T,\mathcal O)$ be a topological space and $S\subseteq T$ a dense set in $T$. If $B(x)$ is a neighbourhood basis of the point ...
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28 views

When a sort of weak topology is enough to generate vector space topology

Consider a vector space $V$, and some functions $f_\alpha: V \rightarrow \mathbb{C}$ where $\alpha$ ranges over some index set $A$. We can think about the coarsest topology which: a) makes the ...
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557 views

A subset E of $R^n$ is totally bounded if and only if E is bounded

I am studying Compactness in metric space with Gamelin and Greene's Introduction to Topology and am confused about lemma 5.4 in the book. A metric space $X$ is totally bounded if for each $e > 0$, ...
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0answers
117 views

$\ f \colon X \to X $ ,continuous function where X is compact,Hausdorff space.Show $\exists A$ st $f(A) =A$.

Suppose $\ f \colon X \to X $ is a continuous function from a compact,Hausdorff space to itself. Prove that there exists a subspace $A$ such that $f(A) =A$. I came up with an answer based on nets ...
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62 views

How can we characterize all topological groups given $G$?

The idea is that all topologies on G (not necessarily making it a topo group) can be completely specified by a set of functions $F = \{f: G \to G\}$ if you form a basis for the topology like: $B = ...
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117 views

Why is proof of the [topological] closed graph theorem incorrect?

Specifically, the closed graph theorem I am referring to is: Let $f : X \rightarrow Y$ exist and $Y$ be compact and Hausdorff. Then $f$ is continuous if and only if the graph of $f$ denoted by $G_f = ...
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55 views

Comparison of two final topologies

Consider the vector space $F$ of all infinite sequences of reals numbers, such that only finitely many terms of each sequence are nonzero. I recently encountered an exercise where I was required to ...
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135 views

n-Torus with antipodal points identified

If we have n-torus $S^1 \times S^1 \times S^1 \times ....$ n times, and $\mathbb{Z}_2$ acts on this just sending each component of $S^1 \times S^1 \times S^1 \times ....$ to its antipodal. What will ...
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88 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
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87 views

If every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space

Show that if every point $x \in X$ has a neighborhood that is Baire space, then $X$ is a Baire space. (Munkres "Topology", 48.3) Here is what I tried : Let $\{U_n\}_{n \geq 1}$ be a collection of ...
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200 views

Sobolev Spaces separable

How do I demonstrate that the Sobolev spaces $W^{1,\infty}$ is not separable? PS: I know that space $L^{1,\infty}$ is not separable but was unable to use this information.
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74 views

Homeomorphism form $(-1,1)$ to $\mathbb{R}$

I want to show that every open intervall $(a,b)$ is homeomorph to $\mathbb{R}$. On $(a,b)$ I chose the relative topology $\mathcal{T}_{(a,b)}$ and on $\mathbb{R}$ the trivial topology $\mathcal{T}$ ...
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54 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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99 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 ...