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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Simple examples of rings from topology

The ring $C([0,1],\mathbb{R})$ of continuous functions from $[0,1]$ to $\mathbb{R}$ is an interesting example of ring due to its some interesting property (namely, structure of maximal ideals). The ...
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1answer
35 views

The topological space $[0,1]$ is a continuous image of the Cantor space question.

Prove that the topological space $[0,1]$ is a continuous image of the Cantor space $(G,T')$. I know that this means to show there exists a function $$(i) f : (G,T') \rightarrow [0,1]$$ such that $f$ ...
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22 views

Global holomorphic vector field on a two-sphere

I'm sure this question has been asked before... Till today, I thought that one cannot define a global holomorphic vector field on a two-sphere due to the hairy ball theorem. However, here's an ...
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1answer
33 views

existence of certain function on unit interval

I'm trying to solve this exercise in an introductory book on general topology: Let $(X,d)$ be a metric space and $A,B \subset X$ disjoint closed subsets. Show that there exists a continuous function $...
5
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1answer
65 views

Complement of a simply connected set is simply connected

I saw the following surprising statement in Wikipedia: When $D\subseteq\Bbb C$ is a simply connected compact set, then its complement $E=D^c$ is a simply connected domain in the Riemann sphere ...
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2answers
28 views

Does this system of open sets have to cover the whole space?

I have been studying basics of descriptive set theory lately. In the lecture notes I follow (sadly, the notes are written in Czech), there is the following definition: Let X be a topological space....
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1answer
47 views

Arc connectedness of Telophase topology

In Counterexamples in Topology Book by Lynn Steen i found that Telophase topology is arc connected. How can we build an arc in this topology ? And also could you give me an idea of how to prove that ...
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2answers
42 views

Proving these equivalent conditions for an open map using boundary of a set

Let $X,Y$ be topological spaces. Prove the following statements are equivalent. $(1)$ $f\colon X\to Y$ is an open map. $(2)$ For all $x\in X$ and open set $U \ni x$ there exists open set $V$...
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1answer
79 views

Given a special topology, how do compact sets look like

I came across the following example of a topology on $\mathbb Z$: $$\mathcal T_{\mathbb Z} := \{M\in \mathcal P(\mathbb Z): M = \emptyset \quad\text{or}\quad M = \mathbb Z\quad\text{or}\quad (-1 \...
5
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3answers
144 views

Isometry map on a compact metric space

Let $X$ be a compact metric space and $f : X\rightarrow X$ such that $d (x,y)\le d (f(x),f(y))$ for all $x,y\in X$. Prove that $f$ is an isometry. I am getting stuck on this question. Can any one help ...
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1answer
35 views

Partition of unity from RCA Rudin

Let me ask the following question: How Rudin applies Theorem 2.7 in the begining? He take some $x\in K$ then $x\in V_i$ where $i=i(x)$. What's next? I thought on this about couple hours but no ...
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1answer
25 views

If a product space is locally compact, then each space is locally compact and all but a finite number of factors are compact

If $\prod^{\infty}_{i=1} (X_i, T_i)$ is locally compact, then each $(X_i, T_i)$ is locally compact and all but a finite number of $(X_i, T_i)$ are compact. Let $X=\prod^{\infty}_{i=1} (X_i, T_i)$, ...
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1answer
36 views

Classification of an open set in real

Prove that open set in real line can be represented as ar most countable disjoint union of open intervals. I know that this question repeated many times in MSE but let me ask the following question. ...
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1answer
19 views

Determine the closure, interior and boundary of the set

What does it mean when it asks for the interior of the set? Also to check, I think this set is open with the boundary at x=0 and an open disk with a radius 1. Am I correct?
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0answers
60 views

Urysohn's Lemma from RCA Rudin

I found out the proof of Urysohn's Lemma from Rudin's book but I have couple questions which I am not able to answer. 1) Why Rudin wrote that "in terms of characteristic functions, the conclusion ...
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1answer
62 views

Subfields of $\mathbb{C}$ which are connected with induced topology

The ring of continuous functions on $[0,1]$ to $\mathbb{R}$ has an interesting property: every maximal ideal of this ring is the subset of all functions vanishing at a common point. If we ...
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19 views

Is $Y/K$ homeomorphic to $Y'$ as defined below -

Let $G$ be a topological group acting on a topological space $X$ in such a way that there are only finitely many orbits. We will fix points $x_1,\cdots,x_n\in X$ and let $X=\bigcup_{i=1}^n G\cdot x_i$ ...
3
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1answer
49 views

can we find a continuous surjection from $\mathbb{R} \to \mathbb{R}^{\omega}$?

I've shown there exist continuous onto map from $\mathbb{R}$ to $\mathbb{R}^{n}$ for any finite $n$. Now my question can we find a continuous surjection from $\mathbb{R} \to \mathbb{R}^{\omega}$ ? ...
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1answer
59 views

Is the complement of the closed unit disk in the plane homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ? [closed]

Is $\mathbb R^2 \setminus D^2$ , where $D^2=B[0;1]$ is the closed unit disk , homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ?
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1answer
63 views

Prove Theorem 2.30 from baby Rudin.

I want to prove that Suppose $Y\subset X$. A subset $E$ of $Y$ is open relative to $Y$ if and only if $E=Y\cap G$ for some open subset $G$ of $X$. The following statement is a bit confusing to ...
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5answers
698 views

Is the plane minus a line segment homeomorphic with punctured plane?

Is $\mathbb R^2$ minus a line segment i.e. $\mathbb R^2 \setminus ([0,1]\times \{0\}) $ homeomorphic with a punctured plane $\mathbb R^2\setminus \{(0,0)\}$ ?
2
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1answer
95 views

Generators of the fundamental groups of the 8-figure and the torus

I have two doubts strictly related to each other. 1) Firstly, consider the $8$-figure, namely the union of two circles in a point $x_1$. Using the Seifert-Van Kampen's theorem I proved that its ...
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0answers
94 views

“admissible” maps from context

I have been reading Massey's Algebraic Topology and on page 158 came across the following "semi-mystical principle" which he says guides much mathematical research: Whenever we wish to gain ...
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1answer
32 views

Inverse function on product topology (Munkres)

I have a simple question that comes from Munkres section 19, Example 2. Let $f:\mathbb{R}\rightarrow\mathbb{R}^{\omega}$ be given by $f(x)=(x,x,x,...)$, with $\mathbb{R}^{\omega}$ a countably ...
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2answers
50 views

Proof verification: diam(E) = diam(closure(E))

Since $E \subseteq cl(E) $, then it is immediate that diam $(E) \leq $ diam(cl($E))$. I only need to show that assuming diam $(E) < $ diam(cl($E))$ will lead to contradiction then I can conclude ...
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1answer
30 views

Can someone please offer a simple definition of “derived net”?

I was looking up a term called "derived net", however the Google search seems to conflate "net" with .NET programming language. (And filter with electronic filters, and "derived net from a filter" ...
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1answer
43 views

Range of $f\in C_c(X)$ is compact subset of complex plane

The collection of all continuous complex functions on $X$ whose support is compact is denoted by $C_c(X)$. Book's proof is quite not detailed and I will write a detailed proof. Proof: Let $f\in ...
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0answers
33 views

Open sets in a discrete product space.

How is it possible for a countably infinite product space to be discrete? If we let each $(X_i,T_i)$ be topological spaces with more than $1$ point and $$\prod^{\infty}_{i=1}(X_i,T_i)$$ be the ...
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0answers
159 views

Deformation retract of wedge sum

Let $(X_\gamma)_{\gamma \in \Gamma}$ be a collection of topological spaces, and let $x_\gamma \in X_\gamma$ be a fixed point for each $\gamma$. Fix some $\alpha \in \Gamma$, and suppose that for $\...
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1answer
14 views

Show that if $x$ is an accumulation point of an ultrafilter on $X$, then the neighborhood filter is contained in the filter

Show that if $x$ is an accumulation point of an ultrafilter $\mathcal{F}$ on $X$, then the neighborhood filter $\mathcal{F}_x$ is contained in the filter i.e. $\mathcal{F}_x \subseteq \mathcal{F}$ ...
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1answer
49 views

Characterizing spaces with no nontrivial covers

I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?
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1answer
35 views

In $T1$ space, all singleton sets are closed?

The definition of $T1$-Space is: A topological space $X$ is said to be $T1$ if for each pair of distinct points $a,b, $ $\exists$ open sets $U,V$ s.t $a\in U, b\notin U, a\notin V, b\in V$. What ...
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1answer
31 views

Random set of rationals topological properties

Flip a coin (probability of heads is p, strictly greater than 0 and strictly less than 1) for every rational number. For each toss, if heads include the number in a set S, if tails exclude it. What is ...
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3answers
42 views

How can a countably infinite product space be discrete?

Let $(X_i,T_i), i \in \Bbb N$, be a countably infinite family of topological spaces. Prove that $\prod^{\infty}_{i=1} (X_i,T_i)$ is a discrete space iff each $(X_i, T_1)$ is discrete and all but a ...
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0answers
32 views

Show that $\mathbb{Q}$ with the topology induced by the Sorgenfrey line is normal

Show that $\mathbb{Q}$ with the topology induced by the Sorgenfrey line is normal. Here, the Sorgenfrey line is the line with the topology whose basis are formed as: $$\{[a,b) : a < b \}$$ I ...
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1answer
40 views

On the matter ; If $f:X \to Y$ is a function with closed graph and compactness preserving then $f$ is continuous

Let $X,Y$ be metric spaces , $f:X \to Y$ be a function , with closed graph , carrying compact sets to compact sets ; then I claim that $f$ is continuous Proof: Let , if possible , $f$ be not ...
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1answer
60 views

Help proving or disproving the following

Let $X,Y$ be topological spaces. Suppose $X=\bigcup_{\alpha\in\Lambda}A_\alpha$ for $\{A_\alpha\}_{\alpha\in\Lambda}$ closed in $X$, then Find a function $f:X\to Y$ such that for all $\alpha\in\...
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0answers
19 views

$f$ is of Baire class $\xi$ implies existence of a topology such that $f$ is continuous with respect to that topology

Suppose that $(X,\tau)$ are $Y$ are Polish spaces and a function $f:X\rightarrow Y$. Show that $f$ is of Baire class $\xi$ if and only if there is a Polish topology $\tau^{\prime} \supset \tau$ with $\...
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1answer
98 views

Infimum of lower semicontinuous functions

The following proposition is from the book Nicolae Dinculeanu Integration on Locally Compact Spaces: Let $H$ and $K$ be two compact Hausdorff spaces and $\alpha$ a continuous mapping of $H$ onto $K$. ...
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2answers
46 views

(Real Analysis) Topology: Prove $f(cl S)\subseteq clf(S)$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Show: $f(\overline{S})\subseteq \overline{f(S)}$ for $S\subseteq \mathbb{R}$ (Note: $\overline{S}$ denotes the closure of S; $\partial S$ ...
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1answer
38 views

Hausdorff & locally compact spaces

1) Every compact space is locally compact. 2) Every metric space is a Hausdorff space. 3) $\mathbb{R}^n$ is a locally compact space. Proof: 1) Suppose $X$ - compact space. Taking $p\in X$ we see ...
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1answer
28 views

Confusing moment from Theorem 2.7 Rudin RCA

I am working with theorem 2.7 from Rudin's RCA book and one moment worries me. Here Rudin uses theorem 2.5 which I added above. Using theorem 2.5 we get that $p \notin W_p$. But how he concludes ...
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1answer
45 views

(Conceptual) Continuity of binary relation $\succsim$ and definition using contour sets

Some background information: $\succsim$ is a binary relation that represents preference between two goods. $\succsim$ means "x is at least as good as y." Continuity of this relation is defined to be ...
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74 views

“Topological” spaces without axioms

A topological space is an ordered pair $(X,\tau)$ where $X$ is a set of points and $\tau$ is a collection of subsets of $X$, satisfying certain axioms (an arbitrary union of things in $\tau$ is in $\...
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about minimal point in non- autonomous discrete system

Let $(X,d)$ be a compact metric space. In $(X,f)$, $x\in X$ is called minimal point if $N(x,U)=\{n|f^{n}(x)\in U\}$ is syndetic for every open set $U$ of $x$ i.e. there is $k\in N$ such that $\forall ...
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23 views

If each $(X_i,T_i)$ is finite and discrete, then the box product is not compact.

If each $(X_i,T_i)$ is finite and discrete, then the box product is not compact. Let each $(X_i,T_i) = \{x_i\}$, then the box product is only covered by the open set $U = \prod \{x_i\}$, but since ...
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0answers
24 views

Convert Baire class $\xi$ function into Baire class one

Let $f$ be a real-valued baire class $\xi$ function. In this paper, page $24$, section $5$, before remark $5.1$, the author defined the set $T_{f,\xi}=\{ \tau^{\prime} : \tau^{\prime} \supseteq \...
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2answers
63 views

Is the empty set a topological space? [duplicate]

If so, is the empty function from it to any other space considered a continuous function? I can't really convince myself either way.
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1answer
30 views

Question about the definition of an open map

Let $X, Y$ be topological spaces. Let $f: X \rightarrow Y$ be a map. If $U \subset X$ is open $\implies$ $f(U) \subset Y$ open in $Y$, then $f$ is an open map. If we have $U \subset X$ open in $X$ $\...
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2answers
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An element is in the cantor set iff it can be written in ternary form with $a_n \neq 1$, for all $n \in \Bbb N$

An element of $[0,1]$ is in the cantor set iff it can be written in ternary form (base $3$) $(0.a_1 a_2 ... a_n ...)$ with $a_n \neq 1$, for all $n \in \Bbb N$. How is this possible? The book I'm ...