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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2
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66 views

Showing that the image of a polynomial map is not closed

Let $f : \mathbb{C}^3 \rightarrow \mathbb{C}^4$ be defined by $(s, t, u) \rightarrow (st, st^2+(1-s)u, st^3, 1-s)$, where $\mathbb{C}$ denotes the complex numbers. Then for some irreducible ...
2
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98 views

Lattice Version of Stone-Weierstrass

I've been reviewing Stone-Weierstrass theoerem. While reading the wikipedia page I read the following version of the theorem: Suppose $X$ is a compact Hausdorff space with at least two points and $L$ ...
2
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45 views

finite simplicial complex compact

Let $K=(V,\Sigma)$ be a finite simplicial complex. I want to show that $|K|$ is compact. I know that $K$ is a sub-simplicial complex of $\Delta^V$ with $|\Delta^V|$ compact. So I think I should show ...
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29 views

A question about the dimension of topological products

For each positive integer n, is the (small inductive) dimension of the topological product of n copies of the "long line", always equal to n? I ask because the "long line" is not a separable metric ...
2
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59 views

Operations on a smooth vector bundle

On a smooth vector bundle, one often defines addition and scalar multiplication to form a vector space. However, doesn't one need to show that these operations are smooth? Is this trivial or is there ...
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28 views

Bounded uniform space

I studied that we do have a concept of total boundedness in a uniform space. I was thinking whether we have a concept of boundedness also in a uniform space (that need not be a metric space). Can ...
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60 views

Topology Bases and Real Numbers

Let $(X,\tau)$ be a topological space. Suppose that $\mathcal{C}$ is a subset of $\tau$, and for every $U$ in $\tau$ and every $x$ in $U$ there exists a $C$ in $\mathcal{C}$ such that $x \in C \...
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62 views

Find $A^\circ , \, cl (A),\, A', \partial A$

Consider the set $$Α=\left\{ \left(\dfrac 1 n, \dfrac 1m \right):\, n,m \in \mathbb N\right\}.$$ We want to find the sets $A^\circ=int\, A, \,cl(A) ,\, A' (\text {= derived set}) , \, \partial A $. ...
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60 views

Complement of contractible subset of a sphere

Let $A$ be a nice closed subset of the sphere $S^n$; for example, we could ask $A\to S^n$ to be a cofibration. Assume that $A$ is contractible. Is then $S^n - A$ also contractible? It ...
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244 views

Using Baire Category Theorem to prove $\mathbb{R}^2\not\cong\mathbb{R}^3$.

How can we prove $\mathbb R ^ 2$ is not homeomorphic to $\mathbb R ^3$ using Baire Category Theorem? Here is a standard proof of this fact using algebraic topology. Note that $\mathbb{R}^{3}-\{x\}$ ...
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119 views

A question about the Skorokhod topology

I have a question which may be naive but I can not find the answer in general reference about Skorokhod topology. Let $\{w_n\}_{n\ge 0}$ be a sequence of cadlag functions defined on $[0,1]$ such ...
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96 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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41 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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81 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 ...
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83 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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768 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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76 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 \operatorname{Int}(A_\...
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40 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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78 views

Problem about compact subspace of Hilbert cube.

This is my problem: I have already completed part (i), but I really can't see how I can relate compact subspace with homeomorphism in part (ii). Please give me some ideas.
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176 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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93 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 path-connected ...
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98 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: Let ...
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54 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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554 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 $...
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33 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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43 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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177 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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78 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 $\...
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54 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: $(1,0),(1,1),(0,1),(...
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175 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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52 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 non-...
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64 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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60 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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59 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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27 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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38 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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68 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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63 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 ...
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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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266 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often leads ...
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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?
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77 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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48 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 $x\...
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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 $f_\...
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621 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 $e > 0$, ...