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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81 views

What does it mean for a topological space $X$ to have a binary open cover?

Does someone know what's meant by a binary open cover of a topological space $X$? I can't find this definition of binary open cover. Could someone who knows it tell me? Thanks ahead for any help:)
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3answers
271 views

compact metric space problem

Let $(X,\tau)$ a compact metric space and $\{ U_i : i \in I \}$ an open cover of X. Show that there is $r>0$ such that for all $a \in X$ there is an $i \in I$ such that $B_{r}(x) \subseteq U_{i}$. ...
4
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2answers
329 views

How much Set Theory before Topology?

I was reading Baby Rudin for Real Analysis and wanted to explore Topology a little deeper. I bought George Simmons' Introduction to Topology and Modern Analysis and found myself liking it. I am having ...
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3answers
594 views

Is there a difference between allowing only countable unions/intersections, and allowing arbitrary (possibly uncountable) unions/intersections?

As in the title, I am asking if there is a difference between allowing set-theoretic operations over arbitrarily many sets, and restricting to only countably many sets. For example, the standard ...
3
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0answers
491 views

When is the closure of a path connected set also path connected?

What are the most general criteria we can impose on a locally path connected Hausdorff space $X$ and a path connected subset $A$ such that $\overline{A}$ is path connected? Do more restrictions need ...
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0answers
62 views

Classes of compact spaces

Given a class $\mathcal{C}$ of compact Hausdorff spaces which is closed under countable products and continuous images. Let $\kappa>\omega$ be a cardinal number. Consider the class ...
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1answer
81 views

Is this space paracompact?

Is this space $F[R]$ with the Pixley-Roy topology paracompact? In general, when the space $F[X]$ is paracompact for general topological space? Definition of Pixley-Roy topology: Basic ...
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2answers
376 views

Topology - The arbitrary union axiom

So, the common answer to why we need the concept of topology is that we need it to talk about things like limits of infinite sequences and continuity. But, when we define the axioms of topology, we ...
2
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1answer
215 views

On compact space with order topology

We are familar with that for the first uncountable cardinality $\omega_1$, the topological space $[0,\omega_1]$ is compact. I find the proof for the $\omega_1$, is also for every regular cardinality. ...
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2answers
714 views

What Topology does a Straight Line in the Plane Inherit as a Subspace of $\mathbb{R_l} \times \mathbb{R}$ and of $\mathbb{R_l} \times \mathbb{R_l}$

Given a straight line in the plane, what topology does this straight line inherit as a subspace of $\mathbb{R_l} \times \mathbb{R}$ and as a subspace of $\mathbb{R_l} \times \mathbb{R_l}$, where ...
5
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2answers
783 views

Showing that the product and metric topology on $\mathbb{R}^n$ are equivalent

I'm new to topology, and can't figure out why the metric and product topologies over $\mathbb{R}^n$ are equivalent. Could someone please show me how to prove this?
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2answers
912 views

Topological Proof that every Interval $I \subset \mathbb{R}$ is connected

First, the definition of connected set: Definition: A topological space is connected iff it cannot be divided in two nonempty, open and disjoint subsets, or, similary, if the empty set and the ...
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2answers
148 views

I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed.

Let $x_n$ be a sequence in a Hilbert space such that $\left\Vert x_n \right\Vert=1$ and $ \langle x_n,\ x_m \rangle =0 $, for all $n \neq m$. Let $ K= \{ x_n/ n : n \in \mathbb{N} \} \cup \{0\} $. ...
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1answer
177 views

Two questions on Sorgenfrey line

Show that the topology of the Sorgenfrey line can be generated be a family of mappings into a two-point descrete space. Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$ but cannot be ...
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0answers
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 ...
6
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1answer
137 views

Decomposing a Polish space into closed sets

Is it possible to decompose $\mathbf R$, or in general, any uncountable Polish space, into $\mathfrak c$-many disjoint closed subsets such that the union of any infinite subfamily is dense? If it is, ...
1
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1answer
106 views

total spaces of S3 bundles over S4 which are homotopic to S7

Milnor showed that if the Euler class of an $S^3$ bundle over $S^4$ is $\pm 1$, then the total space is a homotopy sphere. How many $S^3$ bundles over $S^4$ do we have with the total space is ...
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1answer
2k views

Accumulation points / Cluster points / Closed sets

In a topological space $X$, call $x\in X$ an accumulation point if $\forall$ open set $U\ni x$, $U \cap A \neq \emptyset$, and $y\in X$ a cluster point if $\forall$ open set $U\ni y$, $U\cap ...
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5answers
2k views

Why Zariski topology?

Why in algebraic geometry we usually consider the Zariski topology on $\mathbb A^n_k$? Ultimately it seems a not very interesting topology, infact the open sets are very large and it doesn't satisfy ...
2
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1answer
212 views

Density character of a subspace of a topological space.

Let $(X,\tau)$ be a topological space. Suppose $dc(X)=\kappa$ and let $D\subset_{dense} X$ be a dense subset of $X$ of cardinality $\kappa$. Is it true that $X\setminus D$ has density character ...
21
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1answer
511 views

An interesting topological space with $4$ elements

There is an interesting topological space $X$ with just four elements $\eta,\eta',x,x'$ whose nontrivial open subsets are $\{\eta\},\{\eta'\},\{\eta,\eta'\}, \{\eta,x,\eta'\}, \{\eta,x',\eta'\}$. This ...
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1answer
87 views

Looking for some simple topology spaces such that $nw(X)\le\omega$ and $|X|>2^\omega$

I believe there are some topology spaces which satisfying the network weight is less than $\omega$, and its cardinality is more than $2^\omega$ (not equal to $2^\omega$), even much larger. Network: ...
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1answer
251 views

A universal property for the subspace topology

Let $X$ be topological space and $Y$ be a subset of $X$ with $i\colon Y\to X$ the inclusion map. Show that the induced topology of $Y$ is characterized by the following property: A function $f\colon Z ...
5
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1answer
150 views

Entropy of a Linear Toral Automorphism

I'm trying to calculate the entropy of the Linear Toral Automorphism induced by $$f(x,y,z)=(x,y+x,y+z)$$ This is an exercise in the Katok book. This map has all eigenvalues ​​equal to 1. But I do ...
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1answer
51 views

$U\subset{R}^m$ open $f\colon U\to N$ local homeo and $y\in N$ with $\operatorname{card}\big(f^{-1}(\{y\})\big)$ is infinite then $f$ is not closed.

Let $U$ be an open set. If $f\colon U\subset{R}^m\to N$ is a local homeomorphism and exists $y\in N$ such that $\operatorname{card}\big(f^{-1}(\{y\})\big)$ is infinite prove that $f$ is not closed. ...
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2answers
56 views

Applying a contraction to balls' centers increases the size of the balls' intersection?

The following statement seems clearly true, but I'm having a hard time proving it: Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. For $r\ge 0$, let $B(c,r)\equiv[c-r,c+r]$. Fix ...
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2answers
249 views

Topology question in Rudin book about example 2.21

I am taking a math class where the book used is Walter Rudin. I don't understand how the author explain that in $R^2$ the complex number such $|z|<1$ is not closed open not perfect bounded My ...
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1answer
86 views

a question on Pixley-Roy topology

Let $X$ be a $T_1$ space and let $F[X]$ be $\{x\subset X:\text{is finite}\}$ with Pixley-Roy topology. If $X$ is not discrete, how to proof $F[X]$ is not a Baire space? Thanks ahead:) Definition ...
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1answer
519 views

Continuous functions in product topology

There is a nice theorem characterizing continuous functions $A \to Y^X$ where $Y^X$ is equipped with product topology (pointwise convergence topology). Is there a similar theorem for functions of the ...
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2answers
523 views

Topology for beginners [duplicate]

Possible Duplicate: best book for topology? Please Suggest some good books on Topology and Functional Analysis. It would be good if somebody can post links of video lectures related to ...
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2answers
159 views

On the space of ultrafilters on $N$

I meet the space $X$ of ultrafilters on $N$ with the topology generated by sets of the form $\{p\}\cup A$ where $A\in p \in X$. I can't understand the definition of the topology. Is the points in $N$ ...
2
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7answers
565 views

Best possible question in topology for a beginner to tackle

I know pretty much nothing about topology except that there are closed loops on a torus's surface that can't be continuously deformed into a point. What is the best possible accessible question for a ...
5
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1answer
849 views

Countably Compact vs Compact vs Finite Intersection Property

There is this exercise: Show that countable compactness is equivalent to the following condition. If ${C_n}$ is a countable collection of closed sets in S satisfying the finite intersection ...
3
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1answer
189 views

Is there a topology on the countable set which makes the space is not first countable but has countable pseudocharacter?

I want to know is there a topology on the countable set which makes the space is not first countable but has countable pseudocharacter? Thanks for any help:)
3
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0answers
65 views

Applications of Bitopological spaces [duplicate]

Possible Duplicate: Where do bitopological spaces naturally occur? Do they have applications? J.C.Kelly introduced the idea of bitopological space [Proc. London Math. Soc. (3) 13 (1963) ...
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2answers
4k views

Understanding the definition of a compact set

I just need a bit of help clarifying the definition of a compact set. Let's start with the textbook definition: A set $S$ is called compact if, whenever it is covered by a collection of open sets ...
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2answers
152 views

continuity box topology

Be $x=(x_{n})$,$y=(y_{n})\in \mathbb{R}^\omega$, be $f\colon [0,1]\subseteq\mathbb{R}\rightarrow \mathbb{R}^\omega$ and $f(t)=(1-t)x_{n}+ty_{n}$. For $\mathbb{R}^\omega$ with the box topology, show ...
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2answers
108 views

$\varphi:M\to N$ continuous surjective and closed. Then $f$ continuous iff $f\circ\varphi$ continuous.

$\varphi\colon M\to N$ continuous surjective and closed. Then $f\colon N\to P$ continuous iff $f\circ\varphi\colon M\to P$ is continuous. (Topological spaces) I think that this proposition is ...
3
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1answer
95 views

Topologies for $2^\mathbb{N}$

Let $X = 2^\mathbb{N}$ and $Y = \mathbb{R}^+$ (i.e. the non-negative numbers). Is there a topology in which functions similar to $f : X \to Y$, $$ f(A) = \begin{cases} \frac{1}{|A|}, & |A| < ...
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4answers
149 views

A simple question about open set

Let $X=\mathbb{R}$ and $S=[-1,1]$, i don't quite understand why $[-1,0)$ is open in $S$ but not open in $X$. As far as i know, to show a set is open, we need to show there exist an open ball $B(x,r), ...
2
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2answers
218 views

Does this space have a zeroset diagonal?

Recently I constructed a special topology space which is modifying the example 2.17 of Arhangel'skii as follows : The space is $ Z=X_0\cup X_1\cup X$, where $X_0=\Bbb R\times\{0\}$, $X_1=\Bbb ...
0
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1answer
117 views

Spaces obtained by attaching, as used by Milnor in his book on Morse Theory

In Milnor's book on Morse theory, he argues on page 14-19 for Theorem 3.2. These pages are not available on the google books preview. On p.16 of this argument, he notes, in his notation, that ...
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1answer
154 views

The collection of all compact perfect subsets is $G_\delta$ in the hyperspace of all compact subsets

Let $X$ be metrizable (not necessarily Polish), and consider the hyperspace of all compact subsets of $X$, $K(X)$, endowed with the Vietoris topology (subbasic opens: $\{K\in K(X):K\subset U\}$ and ...
5
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1answer
479 views

Topology needed for differential geometry

I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know? I know some basic concepts reading from ...
5
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4answers
129 views

$f$ closed iff $y\in N$ and open $V\supset f^{-1}\left(\{y\}\right)$ exists $U$ open such that $V\supset f^{-1}(U)\supset f^{-1}(\left\{y\right\})$

Prove that $f\colon M\to N$ (topological spaces) is closed if and only if for all $y\in N$ and all open sets $V\supset f^{-1}\left(\{y\}\right)$ in $M$ there exists an open set $U$ in $N$ containing ...
2
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0answers
114 views

Smooth deformation retracts

Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For ...
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0answers
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 ...
4
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3answers
98 views

Construction of $[a,b]$-fold Cartesian product over space of all real-valued functions

I am currently reading "Applied Analysis", which could be found here, and on page 85 I don't understand example 4.16. There it is said: Suppose that $X$ is the space of all real-valued functions ...
8
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1answer
82 views

Identifying a map by looking at the pair of topologies that makes it continuous.

Let $\omega_X$ be the set of all topologies on $X$. Given $f:X\rightarrow X$, define $R_f \subset \omega_X \times \omega_X $ as those pairs of topologies on $X$ which make $f$ continuous. For example ...
2
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1answer
123 views

Homotopy type of 8-holed torus

I would like to determine the homotopy type of a torus with 8 punctures. (I have come across this problem studying deformations of discontinuous groups of Heisenberg groups...) Other than trying ...