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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18
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1answer
460 views

Is there a homology theory that counts connected components of a space?

It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$. I have recently learned that for Čech homology the ...
13
votes
1answer
180 views

How do Slinkies become tangled?

The following image describes the problem better than I can: As you know, sometimes Slinkies can twist such that the direction of the coil can be reversed. However, though reversed, the coil still ...
13
votes
1answer
313 views

What are the attaching maps for the real Grassmannian?

The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition. The study of characteristic classes tells us that these ...
0
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0answers
221 views

A question about the problem in Functional analysis ( Rudin)

Problem 10( chapter 1, p.39). $ X, Y$: topological vector spaces, $dimY<\infty$, $f:X\rightarrow Y$ is linear, and $f(X)=Y$. (a) Prove that $f$ is an open mapping. (b) Assume, in addition, that ...
0
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0answers
93 views

isolated point of any subspace

If $x$ is isolated point of any $Y\subset X$, there exists $T$ open in Y such that $\{x\}=T\cap Y$. So, $x$ is also isolated point in $X$. Because, $T=U\cap Y$ where $U$ is open in $X$. Then $U\cap ...
0
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0answers
50 views

Lipschitz of multifunction of F

Let $F$ be a Lipschitz continuous multifunction from $\mathbb{R}^n$ to $\mathbb{R}^n$ with the Lipschitz constant $K$ and $$H=\sup\{\langle v,p \rangle|\thinspace v \in F(x)\}.$$ Prove that $K|p|$ is ...
0
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0answers
67 views

How do I prove that $(c,\infty]$ is open in $\overline{\mathbb{R}}$?

$\overline{\mathbb{R}}$ is a topological space, but not a metric space, so I'm not sure if it is true. Let $U$ be an open set in $\overline{\mathbb{R}}$ such that $\infty\in U$. Then, how do I that ...
0
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0answers
149 views

Understanding topologies in dual and bidual.

I don't know if this is a dumb question but i think i better ask and get my confusion clarified . Talking about topologies in a Vector space , Topologies induced by norms are pretty easy to ...
0
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0answers
35 views

Complements and the property of Baire

I am having a hard time understanding why the set of sets with property of Baire is closed under complements. I have read a couple proofs, but they seem to start with a set $B$ with the property of ...
0
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0answers
175 views

A homeomorphism from a semidisk to the unit disk

In Bak and Newman's Complex Analysis Chapter 14, Problem 12, the reader is asked to find a conformal mapping from the upper semidisk (with norm 1) $S$ to the unit disk $U$. Then, they ask to show that ...
0
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0answers
28 views

Regular pseudometrics

Note that the function $x$ to $d(x,A)$ defined on a pseudometric space is uniformly continuous. How does this uniform continuity in a pseudo metric space relate to the regularity of the space?
0
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0answers
92 views

pseudo metric space-regularity

Can somebody please help me to prove that every Pseudo metric space is not regular? Pseudo metric spaces are not Hausdorff.Do we need to use that property here?
0
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0answers
64 views

Surface of a 2-sphere expressed as union of two closed disks

I'm reading a First Course in Differential Geometry by Chuan-Chih Hsiung and on page 8 he says "A closed disk that is homeomorphic to $I^2$ [i.e. $I\times I$, where $I = [a, b]$] is connected. The ...
0
votes
0answers
121 views

Are all even group actions actions on top spaces homeomorphisms?

I'd like to know if an even group action on a topological space is necessarily a homeomorphism. In particular, we say an action $G \times X \to X$ is even if, for any $x \in X$, there is an open ...
0
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0answers
104 views

Is the Hawaiian Earring connected? If yes, is it path connected?

Really stumped on this problem. How should I go about showing that the earrings are connected? Just claim that a circle is connected? To show it is path connected, we need to cook up a function that ...
0
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0answers
96 views

How do I prove this surface is homeomorphic to the sphere

Let $S$ a compact and connected surface. If $S=U_1\cup U_2$, where $U_1,U_2$ are of finite character and the boundary of $U_1$ is in $U_2$. How can I prove that S is homeomorphic to the sphere? I'm ...
0
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0answers
49 views

intuitive idea of deformations in topology

We know that when we prove that two topological spaces are homeomorphic to each other in fact we are proving that these spaces are in fact equal under deformations. Why? this question is very ...
0
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0answers
101 views

stone-cech compactification and sequential space

Let $X$ be a complete regular topologic space and let$\beta X$ denote the Stone-Cech compactification of . Show that every $z\in \beta X\setminus X$ is a limit point of X, but is not the limit of ...
0
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0answers
217 views

Metric Space. Continuity. Compactness.

Let $(X,d)$ be a metric space; let $A$ be a nonempty subset of $X$. For each $x \in X$, we define the distance from $x$ to $A$ by the equation $$d(x,A) = \inf\{d(x,a) | a\in A\}.$$ a) Show that the ...
0
votes
0answers
50 views

functions from the sphere

Can we assign in a continuous manner to each point of the sphere $S^2$ a two point subset of S^2? I think this would contradict in some way "The Poincare theorem" Thanks
0
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0answers
86 views

Is the notation for suspension of topological spaces because of spheres?

It's a fairly basic topological fact that $SS^n = S^{n+1}$ where $SX$ denotes the (free) suspension of $X$, and $S^n$ is the $n$-sphere. Do these notations have to do anything with each other, or is ...
0
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0answers
173 views

Closure and Limit Points

Let E' be the set of all limit points of a set E. Prove that E' is closed. Prove that E and E closure have the same limit points. Do E and E' always have the same limit points? -Thx-
0
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0answers
82 views

G is a topological group acts on topological space $X$, is $f_{g}:X\rightarrow X, x\rightarrow g*x$ continuous?

Let $G$ be a topological group acts on the topological space $X$, for an elememt $g\in G$, let's define the map $f:X\rightarrow X, f(x)=g*x$. I am trying to find if $f$ is continuous? my best ...
0
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0answers
137 views

Finite dimensional normed space

I would like to find an elementary proof of the following theorem Let $E$ be a normed space. Then the following statements are equivalent: (a) E is finite dimensional. (b) Every linear functional ...
0
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0answers
65 views

First examples in triangulations

I am starting to study about triangulations in my algebraic topology course. We have seen the triangulation of the sphere, the closed disc and so on. Intuitively it's ok, however I couldn't find any ...
0
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0answers
77 views

a question on Hopf fibration

While reading a note on hopf fibration we came proving $\mathbb{CP}^1$ is homeomorphic to $\hat{\mathbb{C}}$ author says like $\mathbb{CP}^1$ has the quotient topology $\mathbb{CP}^1=S^3/\sim$. We ...
0
votes
0answers
147 views

Somewhat complicated example demonstrating the failure of the Pasting Lemma?

Suppose we let $A_0 = [0,1]$. Now let $A_1$ be the line segment of length $1$ connecting the origin to the Cartesian coordinate $(0,1)$. Define $A_2, A_3, …$ to be similar line segments each of length ...
0
votes
0answers
77 views

'set of limit points of $E$' and $E$ have the same limit points?

Let $X$ be a metric space and $E\subset X$. Let $E'$ be the set of limit points of $E$. I know that $\overline E$ and $E$ have the same limit points l, but what about $E'$?
0
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0answers
137 views

Embedding of a finite simplicial complex

In the paper Hardness of embedding simplicial complexes in $\mathbb{R}^{d}$ the abstract states that a finite simplicial complex of dimension $k$ embeds in $\mathbb{R}^{2k}$ while on page $856$ he ...
0
votes
0answers
409 views

Every k-cell is compact (in ZF)

This is the part of proof on my book. Let $I$ be a k-cell. Then for every $x\in I$ and $j\in k$, $a_j ≦ x(j) ≦ b_j$ for some $a_j, b_j \in \mathbb{R}$. Let {$G_{\alpha}$} be an open cover of I and ...
0
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0answers
83 views

Algebraic topology involved in the 1/4-pinched sphere theorem?

Can anyone familiar with this theorem and its proof let me know how much algebraic topology is involved, and where specifically? I am familiar with a lot of differential geometry, but not many of the ...
0
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0answers
122 views

An example with a non-increasing function

Let $\mathfrak{A}$ is a poset. For $a, b \in \mathfrak{A}$ we will denote $a \curlyvee b$ if only if there is a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$. Let ...
0
votes
0answers
126 views

Intersection of $n$-connected sets in $\mathbb R^m$

Consider two $n$-connected sets $U,V\in \mathbb{R}^m$. What is the minimal $m$ such there exist $U,V$ such that $U\cap V$ is not connected? $n$-connected?
0
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0answers
129 views

Algebraic varieties and Hausdorff spaces

Let $(X,\mathcal O_X)$ be an algebraic prevariety, by definition, it is an algebraic variety iff the diagonal $\Delta(X)$ is closed in the product $X\times X$. The above property is equivalent to the ...
0
votes
0answers
207 views

Topological product of filters

It is trivial that every set theoretic filter with added empty set is a topology (a collection of open sets). What if we consider products of filters considered as topological spaces? (both Tychonoff ...
0
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0answers
73 views

“componentwise constant”?

This is a trivial vocabulary question. It seems to me that "constant on every connected component of the domain" would be a reasonable definition of the term "componentwise constant", provided that ...
0
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0answers
60 views

Continuity of a map of a topological space to a pro-topological space

Let $(X_i)$ be a projective system of topological spaces. Let $X$ be the projective limit of $X_i$. Let $G$ be a topological space. What does it mean for $G\to X$ to be continuous? My guess is that ...
0
votes
0answers
215 views

Limit of a function as approaching an isolated point of its domain?

From Wikipedia Suppose $X,Y$ are topological spaces with $Y$ a Hausdorff space. Let $p$ be a limit point of $Ω⊆X$, and $L ∈Y$. For a function $f : Ω → Y$, it is said that the limit of $f$ as ...
0
votes
0answers
232 views

Is proving sequential continuity more difficult than proving continuity?

For a function $f: \mathbb{R}^n \to \mathbb{R}^m$, I know that continuity and sequential continuity are equivalent. Sequential continuity of $f$ at a point $x \in \mathbb{R}^n$ means for any sequence ...
0
votes
0answers
45 views

zero set projective space

Here's my question. Let's consider the polynomial $p(y_{1},...,y_{n})$ with $deg(p)=d$. The set $C=\{(y_{1},...,y_{1})\in \mathbb{R}^{n} | p(y_{1},...,y_{n})=0\}$ is closed in $\mathbb{R}^{n}$ ...
0
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0answers
70 views

Separability in ultrametric spaces

A separable, ultrametric space $X$ is given. Does that mean that one immediately gets a countable cover of $X$ consisting of open balls with radius $r>0$ by virtue of separability? (I mean to ...
0
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0answers
70 views

Topological Disconnected Graphs

How can I define a topology on a complete graph such that the connectedness of the subgraphs of it and the connectedness of the sets on the topological space become equivalent?
0
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0answers
261 views

Is an $\omega_1$-Lindelöf space with a $G_\delta$- diagonal a Lindelöf space?

Suppose $X$ is an $\omega_1$-Lindelöf space with a $G_\delta$-diagonal. I want to show it is a Lindelöf space by the method of Chaber from 1976, by which he proved a countably compact space with a ...
0
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0answers
59 views

Pointed bundles as short exact sequences

Let $\pi: E \to B$ be a pointed continuous surjection, and let $F = \pi^{-1}(b_0)$ be the fiber over the basepoint (base fiber for short). Then $$* \to F \to E \stackrel{\pi}{\to} B \to *$$ is a short ...
0
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0answers
43 views

techniques of construction of functions using power series and uniform convergence

I´ll refering to "continuous" functions $$ f:R \to R $$ with it´s usual topology I've heard that using power series, can be simplified a lot of work to build a certain type of functions, like using ...
0
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0answers
58 views

connectedness of a special class of closed curves in $\mathbb{R}^{3}$

during some geometrical speculations I came across the next problem: Let $\mathcal{P}$ be the class of all continuous spatial closed curves $r:\mathbb{S}^{1}\rightarrow\mathbb{R}^{3}$, in coordinates ...
-1
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0answers
41 views

Is $(\mathbb{R},+)$ a smooth manifold?

I feel like $(\mathbb{R},+)$ is, but I'm not really sure. How would I know whether or not it is?
-1
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0answers
88 views

mathematical economics (topology)

Let $I$ be a finite set of players and let $X=\{x_1,x_2,\ldots,x_n\}$ be the state space. let $A_i$ be player $i$'s action space and let $A = \prod_{i \in I} A_i$. suppose $s_0$ belonging to $X$ is ...
-1
votes
0answers
34 views

show that euler characteristic of order complex is zero

Let P be the poset of all non empty subset of {1, 2, .....,n} under set inclusion. Show that euler characteristic of order complex deltaP is zero I try induction...but failed..
-2
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0answers
31 views

Real analysis and topology

Let $(X,\tau)$ be Hausdorff topological spaces, Show that $\tau$ is a semi-ring if and only if $\tau$ is the discrete topology.