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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31 views

Why use class multiplication in Homotopy groups?

This is a related to a physics question Why use class multiplication to describe topological entangling and merging?. In physics, the homotopy theory is used to describing topological defects in order ...
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29 views

Homology of Subspace vs. Homology of Ambient Space.

Let $M$ be a manifold embedded in $\mathbb R^n$ , so that the manifold has non-trivial $k-th$ homology for some $n \geq k\geq 0$ . How do we identify the fact that while there is a non-trivial cycle ...
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12 views

Chain boundary (Topology)

Could anyone give me a topological definition of chain boundary, if possible one which could be integrated in further definitions (homology, quiver, bound chain and so on)??
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71 views

The triangle inequality for shortest paths of graphs

In why-the-triangle-inequality I found the statement: for example if $d(a,b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite ...
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38 views

“Absolute retracts” in arbitrary category

Is there a standard notion of something like "absolute retract" in arbitrary categories that generalizes absolute retracts in topology? I am mostly interested in categorical approach to Hausdorff ...
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17 views

Continuous scalar field at an interior point of S and same sign proof.

Let $f$ be a scalar field continuous at an interior point $a$ of a set $S \in R$. If $f(a)$ is not $0$, prove that there is an $n$-ball $B(a)$ in which $f$ has the same sign as $f(a)$. The above ...
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82 views

Cantor set--nowhere dense, complete

I can't figure out this out. Cantor set is closed in $\mathbb{R}$. $\mathbb{R}$ is a complete metric space. Every closed subset of a complete space is also complete; thus, so is the Cantor set. ...
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47 views

How to convert an object into a sphere?

I'm not sure if I understand it enough, but doesn't the Poincare conjecture prove any shape can be turned into a sphere? How would I go about transforming such an object? Like let's say I have a ...
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43 views

Any limit definition defines topology

Let $X$ be a normed space and let $C(X)$ be the space of the continuous functions $X\to R$. We have the dual space $C\left(X\right)^{*}$. Suppose we have the definition of limit on the dual space. ...
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38 views

What's the link between a pseudo-topology and a topology?

In distribution theory one considers, amongst other things, the space $\mathcal{D}(\Omega)$. In some French books, it is said that we confer a "pseudo-topology" to $\mathcal{D}(\Omega)$ when defining ...
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36 views

Coverings maps of a simply connected space

Let be $Y$ a simply connected space. Show that $Y$ doesn't admit covering maps that aren't homeomorphisms, ie, every cover space of $Y$ is trivial ($I\times Y$, with $I$ a discrete space). So, I know ...
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28 views

To prove existence of an open set of functions

I am trying to prove the following: In $C(X,Y)$ with $X=[0,1]$ and $Y$ of finite dimension $K$, $C(X,Y)$ having the topology of uniform convergence, for any $K$ finite there exists an open set of ...
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51 views

Why $\mathcal{D}(\Omega)$ be a topological vector space is important?

Let $\Omega\subset\mathbb{R}^N$ be an open set and $\mathcal{D}(\Omega)$ the set of all infinitely differentiable functions with compact support on $\Omega$. In the study of PDEs, we use the ...
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21 views

What is the domain and image of the composition of mappers in a manifold

I was trying to understand the following: which I got from: http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf I was wondering, why the domain was: $$ \alpha(U_{\alpha} \cap ...
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31 views

Contractibility of complex manifold

I'm trying to show that for $v^2 = w^4 - a^4$ for real $a$ and complex $v, w$ that this manifold deform retracts to a point $(0, a)$ but can't seem to figure out a path that remains on the surface. ...
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23 views

Finding a covering map of a complex manifold

Is there an easy way to find a covering map of a complex manifold given by $(w, z) = 0$, where $w, z$ are complex numbers?
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18 views

Give a basis for the countable topology on X$^w$

Define the $\ell$$^2$-topology on X to be given by the metric d = [$\sum_{i=1}^\infty$(x$_i$ - y$_i$)$^2$]$^\frac 12$ Give a basis for the countable topology on X$^w$. Will the basis for X$^w$ be ...
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35 views

How do we call such a compactification?

Let $E$ be a denumerable set and let $\mathcal{F}$ be a collection of bounded functions. In the reading we had a compactification of $E$ with respect to $\mathcal{F}$, denoted by ...
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38 views

Question about Open Covers

I am trying to find an open cover for an interval $I = [0,2]$ such that the open sets that cover this interval have a $20\%$ overlap. For example: I have the interval $I=[0,2]$. I want the length of ...
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24 views

How to form a surface with two cross caps by attaching a Mobius Band

I'm trying to find a way to realise the surface with two cross caps as identifications on a hexagon. To do this I'm trying to attach a Mobius band to the boundary of a circle removed from the real ...
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44 views

$f:[0,1]\rightarrow\mathbb{R}$. For every $x\in[0,1]$ exists $n\in\mathbb{N}$ s.t $f^{(n)}(x)=0$ . Prove that $f$ is polynomial.

I proved that every closed interval $[a,b]\subseteq [0,1]$ contains an open interval $J$ s.t $f|_{J}$ is polynomial: Let $[a,b]\subseteq [0,1]$. Consider $A_{n}=\{x\in [a,b] \mid f^{(n)}(x)=0\}$. ...
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62 views

Minkowski Sum and subtraction

Consider the following prompt: Prove that $ ( A\oplus B )\ominus$ $B$ and $( A\ominus B)\oplus$ $B$ need not equal $A$ for all sets $A$, $B$,where $\oplus$ and $\ominus$ denote the ...
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14 views

Upper hemicontinuity of a correspondence

I would like to know whether the following correspondence is upper hemicontinuous: $$ C(x)=\begin{cases} 1, & (f(x)>0) \\ [0,1], & (f(x)=0) \\ 0, & (f(x) < 0) \end{cases}, $$ ...
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58 views

What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
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34 views

Which quotient spaces is Hausdorff in Klaus's book?

In Chapter3.3 "Properties of Quotient Spaces" of the book "TOPOLOGY" written by Klaus Janich and translated by Silvio Levy. I can't figure out which one is Hausdorff. Any one can explain it? Update: ...
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17 views

Construction of the completion of a spread

I am reading Fox's paper Covering Spaces with Singularities and am a bit confused in a part where he constructs completions of spreads. The relevant definitions may be found here: A Complete Spread ...
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24 views

Dual of path object

For a topological space X, what might be the dual of the path space $X^I$ of X? Does it make any sense to think of the topological cylinder X x I over X as dual to the path space over X?
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55 views

Why quotient map maps open sets to open sets?

My book says that the quotient map of a topological space $X$ into its equivalence classes under some equivalence relation maps open sets to open sets.However my intuition tells me that this is wrong ...
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39 views

The “retraction” of $S\subseteq\Bbb R^2$ has rectifiable boundary

This is a continuation of the line of investigation of What's the most efficient way to mow a lawn? (although this question is self-contained). For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define ...
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69 views

Showing the sphere is not homeomorphic to a torus (my own question!) (or indeed a circle and a washer) - OR puncturing is not continuous

Motivation imagine a rubber sheet extended over the end of a tube, I am saying: "there is no continuous transformation that can retract this sheet over the side" - it is common place to talk about ...
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42 views

Are the face posets of CW-complexes Eulerian?

Suppose we had a CW-complex $X$ with decomposition $X_{i}$ Is its face poset, consisting of cells and covers generated by the attachment of cells to one another, an Eulerian poset? What would be the ...
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38 views

Homeomorphism between the group of $S(O)_{2}$ and the $S_1$.

During an exam I had to prove the following: "Let there be a dynamical system of $n=2$ dimensions and let the eigenvalues that correspond to it, to be imaginary with their real part equal to zero. ...
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44 views

Homeomorphism between $S^1\times \Bbb{R}$ and $\Bbb{R}^2\setminus\{(0,0)\}$

I need to show that there exists a homeomorphism between $S^1\times \Bbb{R}$ and $\Bbb{R}^2\setminus\{(0,0)\}$ If we map $(\cos x, \sin x,y) \to (y\cos x,y\sin x)$ then only problem is that it is ...
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47 views

Hilbert cube in metric Spaces

Is there proof that does not use the concept of " product space " to prove that the Hilbert cube is closed and bounded ?
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41 views

Induced topology by a complete uniform space.

I know that Uniform space is generalization idea of metric space,Uniform space like metric space induce a topological space. Now my question is ( or are ):- In case our Uniform space was complete ...
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30 views

Question related to Uniform Space

I have questions related to Uniform Space; If $X$ is a countable discrete space, then how to show that finest pre compact uniformity on $X$ admits a countable base of entourages. If $\mho$ is a ...
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18 views

bicontinuous function satisfying certain conditions

Let $(X,\tau_1,\tau_2)$ be a bispace (a space with two topologies $\tau_1$ and $\tau_2$).We say that a function $f: (X, \tau_1, \tau_2) \rightarrow (\mathbb{R}, u, l)$ is bicontinuous if $f : (X, ...
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23 views

If $f$ is a loop in $\mathbb{S}^{n}$, then is $f^{-1}(\{x\})$ a compact set in $[0, 1]$?

If $f$ is a loop in $\mathbb{S}^{n}$, then is $f^{-1}(\{x\})$, $x \in \mathbb{S}^{n}$, a compact set in $[0, 1]$?
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57 views

Induced order topology on subset of $\mathbb{R}$

Let $X = (0,1)_\mathbb{R} \cup [2,3]_\mathbb{R} \cup (4,6]_\mathbb{R} \cup \{8\}$. Suppose $X \subset \mathbb{R}$ is given the Euclidean ordering and the induced order topology. Determine whether ...
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35 views

the motivation of separable field extension

What is the origin of the motivation of separable field extension? Is it related to separable topological space or something else?
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21 views

Showing a map is a bounded linear operator.

Show that the map A : (C[0,1],∥·∥∞) → R, Ax = x(0), ∀x ∈ C[0,1] is a bounded linear operator. I know one has to show the map is continuous but I'm not sure how to go about proving it in this case. ...
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31 views

Any two continuous maps $f,g: X \to B$ that agree on $A$ are homotopic relative to $A$.

Let $B\subset \mathbb{R}^n$ be any convex set, $X$ any topological space, and $A$ be any subset of $X$. Then any two continuous maps $f,g: X \to B$ that agree on $A$ are homotopic relative to $A$. My ...
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36 views

Countable metric space and space of all rational numbers.

I know that uncountable metric space can not be homeomorphic to a subspace of the space of $\mathbb{Q}$ of all rational numbers. but can this be true for countable metric space?
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55 views

Sequential compactness in weak topology

When the Banach space $V^*$ is reflexive, we have the unit ball in $V^*$ is weak$^*$ sequentially compact. For a Banach space $V^*$ that might not be reflexive, we have to assume that $V$ ...
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40 views

Exhaustion of a manifold by compacts

I searched for a proof of the following statement, but did not find one. I want to check if a proof I made is correct, or if I'm leaving out some detail and/or complicating things: Proposition: ...
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22 views

Seperation of convex compact sets with affine halfspaces

Let $C_1,C_2,...,C_m$ compact convex sets s.t. $\bigcap C_i = \emptyset$. I want to show that in that case there exsist affine halfspaces $H_i$, such that for every $i=1,2...,m$, $C_i \subset H_i$ ...
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29 views

Center of real projective line or Riemann sphere

I have recently encountered the ideas of the real projective line and the Riemann sphere, and it seems to me that in any circle (representing the real projective line) or sphere, the center is a ...
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29 views

Classification of surface with 18-gon planar diagram

For starters, this is a problem from L. Christine Kinsey's "Topology of Surfaces." The problem is to classify the surface using cut and paste arguments on polygons. However, between my limited ...
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23 views

$\phi$ is a coboundary iff $\phi(f)$ depends only on the endpoints of $f$

I've proved the first direction but I'm having trouble proving the second direction. First direction: Let $\phi = \delta\psi$ for $\psi \in C^0(X;G)$. Then $\phi(f) = \delta\psi(f) = ...
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25 views

Question on weak star convergence in subspace

Let $X$ be some normed linear space and let $X^\ast$ denote its dual space endowed with the weak star topology. Let $U^\ast$ be some subspace of $X^\ast$. If I want to show that $\varphi_n$ ...