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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Question on homotopy lifting

I'm studying covering maps and homotopy lifting and I would like to clarify a few things which my lecture notes doesn't seem to make clear. A lemma in my lecture notes says: Let $p: \tilde Y \to ...
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83 views

Orientation of the barycentric subdivision

Two orderings of the vertices of an $n$-simplex are said to be equivalent if they differ by an even permutation. An orientation of an $n$-simplex is a choice of one of the two equivalence classes of ...
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63 views

Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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114 views

Find the closure for several sets

(a) $\mathbb{Q}$ (b) {$(x,y)\in\mathbb{R}^2:xy<1$} (c) {$(x,\sin($${1}\over{x}$$)):x>0$} (d) {$(x,y)\in\mathbb{Q}^2:x^2+y^2<1$} First Closure $\overline{A}$, it is a set contains all ...
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72 views

Continuity and openness of the map $C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$

I need to prove or disprove that the composition operator is continuous and open. Consider the following map $$h:C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$$ that takes a function ...
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65 views

Completely Regular Topological Space and Measure Theory

Here is the statement... Suppose that $(X,\tau)$ is a comletely regular topological (I think the lecturer requires X to be Hausdorff too.), and that E is a dense linear subspace of ...
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423 views

Collection of open intervals $(a,b) \ a,b\in \mathbb{Q}$ is a basis for euclidean topology on $\mathbb{R}$

I'm not sure if this question hasn't already been asked here, but I couldn't find it. I'm currently studying topology and I'm reading a book which unfortunately has no answers to the exercises. ...
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35 views

What is the appropriate def. of $\sigma$-($\Sigma^1_1$) measurable.

I know that borel measurable means that the inverse image of a Borel set (or open set) is measurable. Edit: I am speaking of the sigma algebra generated by the analytic sets in a top. space.
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45 views

Sierpinski's triangle set of local cut points

Could you please help me identify Sierpinski's triangle set of local cut points? I know it does not have cut points but it has local cut points.Which are they? Thank's in advance !
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33 views

Polar representation of continuous curve.

Given a continuous curve $\gamma:[a,b]\to\Bbb R^2$ with $\gamma(t)\ne(0,0)^T$, can I always find continous functions $\rho:[a,b]\to\Bbb [0,\infty)$ and $\theta:[a,b]\to\Bbb R$ so that ...
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86 views

Questions on compact space

Let $X$ be compact. Q1, Does there exist a dense subset $Y$ of $X$ such that the tightness of $Y$ is countable? Q2, Does there exist a dense countably compact left-separated subset $Y$ of $X$? ...
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85 views

Analysis and real analysis

Let $X$ and $Y$ be compact metric spaces. Let $X\times Y = \{(x; y) \,:\, x \in X;\, y \in Y \} $be the cartesian product. Show that any $f \in C(X \times Y )$ can be uniformly approximated by ...
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70 views

Why are pointwise and uniform convergences topologizable?

My question is about two types of convergence on functional spaces. Why can pointwise convergence for mappings from a set to a topological space be topologizable? Why can uniform convergence for ...
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88 views

Continuity in the Strong(Whitney) Topology

Let $P,M$ and $N$ are smooth manifolds and let $F:P\times M \to N$ be a smooth map. We know its associated map $\tilde F:P \to C^\infty(M,N)$ given by $p \mapsto F_p(m)$ is continuous if and only if ...
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103 views

2-sphere union with unit 2-cell is homotopy equivalent to one point union of two 2-spheres.

This is problem 2 in Bredon I.14, on homotopy. I need to prove that $X$ = union of the 2-sphere with the unit 2-cell going through the origin is homotopy equivalent to $Y$ = one-point union of two ...
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121 views

Different definitions of regularity of a measure

I was wondering what relations are between these different definitions of a regular measure? When are they equivalent? There are two non-equivalent definitions from Wikipedia Let $(X, T)$ be a ...
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66 views

Fréchet mean for a general shape space

I am posting this question in order to gain a better understand of what the Fréchet mean is for a generalised shape space. So firstly I gather that the Fréchet mean of a probabilty measure $\mu$ on a ...
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499 views

Continuity of a Characteristic function

Let $A$ be a subset of $\mathbb{R}^n$. Show that the characteristic function $\chi_A$ is continuous on the interior of $A$ and on $A^c$ but discontinuous on the boundary of $A$. My attempt: Suppose ...
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100 views

Which space this space drawn in this picture is homeomorphic?

Based in this question Why this space is homeomorphic to the plane? I would like to know which space this space is homeomorphic, any help or an intuitive idea are welcome. [Context of Image: ...
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91 views

Two problems on topological properties of two different sets

Which properties hold for the following sets? Open, connected, closed, nowhere dense? $$A =\{(x,y) \in\mathbb{R}^2 \mid y=mx\}\setminus \{(0,0)\}\subset\mathbb{R}^2$$ $A$ is the ...
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78 views

A theorem of a continuous map $f: S^1 \to S^1$

This is a theorem from my lecture notes: If the continuous map $f: S^1 \to S^1$ extends to a continuous map $F: B(0,1) \to S^1$ the $f$ is homotopic to a constant map. The proof just defines a ...
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64 views

How to prove this isotopy exist?

Let $M$ be a topological manifold and $N$ is a subset of $M$. Let $f_t$ be an isotopy from $id $ to $f$ which is a homeomorphism on $M$. Suppose $f(N)=N$ and there is an isotopy $g_t$ on $N$ such that ...
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55 views

Existence of limit inferior and exterior

First of all, i'm sorry that i don't know what the title should be for this question. Please edit the title if there is a better way to describe this question. ============= Here's a definition from ...
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34 views

Other definitions of singularity

Many definitions of a singularity of a manifold $X^n$ are concentrating on the defining equations of it and the vanishing of the (partial) derivatives. My questions: What if $X^n$ isn't algebraic ...
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58 views

Continuous homomorphisms

I was reading JS Milne's book on Arithmetic duality theorems and he states on page 105 that for a finitely generated torsion-free G-module (G is actually a galois group) M we have ...
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102 views

Questions about $\sigma$-algebra, algebra and topology

I know the definitions of $\sigma$-algebra, algebra and topology, but why countable/finite union, as in $\sigma$-algebra/algebra, and finite intersection, arbitrary union as in topology? What inspire ...
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32 views

$L^2$ of a fiber bundle

For spaces $X$ and $Y$ with measures there is an isomorphism of Hilbert spaces $$L^2(X) \otimes L^2(Y) \to L^2(X\times Y), ~~ f\otimes g\mapsto \left((x,y)\mapsto f(x)g(y)\right).$$ Now suppose $E ...
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98 views

How do I prove this function is not continuous?

Let $\alpha$ a path from $[0,1]$ to a topological space X. Let $\alpha(0)=\alpha(1)=c$, where $c\in X$. The standard function to prove that $\alpha\cdot\bar \alpha$ is homotopic to the constant map ...
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50 views

example of area preserving twist maps

I recently learned about the poincare-birkhoff fixed point theorem, and was wondering if there is a simple example of an area-preserving (or some weaker assumptions) twist map on the annulus which has ...
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76 views

Definition by commutation property on structures : continuity and where?

(This is very vague, so sorry if there are approximations) I remember that one can define continuity as a commutation property of a function with the limit operation. Structurally, i think it maps a ...
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201 views

triangulation of the closed disc

I'm starting to study triangulations of topological spaces by myself. I find really difficult since I've never seen any formal example of a triangulation in any book! So, I began with the one of the ...
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45 views

Manifolds question

Let $M$ subset of $R^{n+p}$ be the zero set of a $C^\infty$ mapping $g:R^{n+p} \rightarrow R^{p}$. Assume that the Jacobi matrix of $g$ has rank $p$ everywhere on $M$. Show that $M$ is an ...
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104 views

The complement of a submanifold in a manifold

Let $X$ be a topological $n$-manifold and $N$ a $d$-submanifold of $X$, ($d\leq n$), then under what conditions on $X$ and $N$ do we have that the complement $X-N$ is again a manifold and what is the ...
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46 views

Sign-preservation of continuous map in a small neighborhood

So I was reading a small book on surfaces called "Mostly Surfaces" which is available for free in the internet: http://www.math.brown.edu/~res/Papers/surfacebook.pdf In page 32, the author decides ...
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72 views

A function has at least 4 critical points on $[0,1]\times[0,1]$

Define a smooth function $f(x,y)$ on $\mathbb{R}^2$,which satisfies the following condition:$$f(x,y)=f(x+1,y);\quad f(x,y)=f(x,y+1)$$ (i.e.$f$ can be defined on a torus)then hotw to proof that there ...
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67 views

Partition of open sets in $\mathbb{R}^d$.

Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets. In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
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60 views

Pointfree generalization of uniform spaces?

Topological spaces generalize as frames and locales. But are there a pointfree generalization of uniform spaces?
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86 views

Neighborhood Retraction of Boundary

Here is the problem: If $M$ is a manifold with boundary, then find a retraction $r:U \rightarrow \partial M$ where $U$ is a neighborhood of $\partial M$. I realize that the collar neighborhood ...
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102 views

Is there a fractal origami shape that trades volume for area to always keep a flat surface when expanded?

I'm thinking of something like a 2.5D sierpienski type shape. The idea is to enable an lcd type screen that could unfold to "any" size by unpacking space filling elements packed in 3d to a 2d ...
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214 views

Identity component of a Lie group

Could any one help me to solve this problem? Let the identity component $G_0$ of a Lie Group $G$ be the connected component of the identity element $e\in G$. Let $\mu$ and $i$ be the multiplication ...
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71 views

Uniform covers and partitions of unity

A cover $\mathcal C$ of a uniform space $(X,\mathcal U)$ is called a uniform cover if there is $U\in \mathcal U$ such that the cover $\{U(x):x\in X\}$ refines $\mathcal C$. Is it true that to every ...
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68 views

Adjoint to the Hom functor in Boolean rigs

What I wanna ask is about analogies to the tensor product for commutative boolean rings. What I mean by commutative boolean ring is set with two operations, + and *, and two identities, 0 and 1, as is ...
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128 views

Notation in Gilbarg/Trudinger? [Section 2.8]

Note: as discussed below, there is no mistake here. The notation and conventions have been cleared up for me. However what I have written is not incorrect, either. At least to me, it is just a ...
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159 views

topological group

Recently I'm interested in this open question: Must every star compact topological group be countably compact? star compactness ( which implies pseudocompactness ) = for every open cover $U$ of ...
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220 views

Properties of Topological Groups

I'm working though William Basener's Topology and Its Applications and I have come across a problem I can't solve. The book defines a topological group as a group equipped with a topology where for ...
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124 views

Countable Product of discrete spaces

Let $X$ be a countable discrete topological space. Consider $X^{\mathbb{N}}$ endowed with the product topology. How do you prove that $X^{\mathbb{N}}$ is homeomorphic to the sub-space of all ...
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90 views

Tangent bundle topology

Suppose $M \subset \mathrm{R}^n$ is a smooth manifold. Does the typical definition of the topology of the tangent bundle $TM$ of $M$ by charts coincide with the topology of $TM$ regarded as a subspace ...
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133 views

Definition of topologically onto map and Covering Spaces

In the chapter on Covering Spaces in his book A Basic Course in Algebraic Topology Massey uses the term topologically onto when defining covering spaces (see below for Massey's definition). What does ...
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164 views

Applications of monads in general topology?

What are applications of monads in general topology? For example, for GT is important the notion of products, products are adjoints, so adjoints may be important for GT, but what's about monads?
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88 views

Question related to some class of nowhere dense sets

Let $\Omega\subset (0,1)$ be a nowhere dense set which has no lower and upper bound in $(0,1)$ and for which $\Omega^{d}\cap(0,1)=\Omega$ ($\Omega^{d}$ denotes here the set of all limit points of ...