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 about continuous onto maps of homeomorphic spaces.

If $f:(A,T) \rightarrow (B,T_1)$ is continuous and onto, and $$(A,T) \cong (C,T_2) \land (B,T_1) \cong (D, T_3)$$ $$\Rightarrow \exists g: (C,T_2) \rightarrow (D,T_3)$$ that is continuous and onto.
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33 views

Function from space of continuous functions to reals is continuous (Proof Verification)

Question: $C$ is the space of continuous functions from $[0,1]$ to $\mathbb{R}$ under the sup metric. Prove the function $$f:C\to\mathbb{R}\quad f\to \int_0^1 f(t)^2 dt$$ is continuous. My answer: ...
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Comparing different definitions of tightness for measures

Let $X$ be a Hausdorff space, $\mathcal{B}(X)$ the Borel $\sigma$-algebra and $\mu : \mathcal{B}(X) \to [0, \infty]$ a measure. Consider the following properties: (1) $\forall A \in \mathcal{B}(X): \...
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1answer
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Question about the composite of a homeomorphism and a continuous onto function.

If $f : (G,T)$ homeomorphically to $(A,T_1)$, and $h: (A,T_1)$ continuously and onto $(C,T_2)$, then is it always the case that, given the composition $g = h \circ f : (G,T) \rightarrow (C,T_3)$, the ...
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2answers
67 views

Are these subsets homeomorphic?

Are the two subsets of the Euclidean Plane $[0,1]\times[0,1)$ and $[0,1)\times[0,1)$ homeomorphic or not? My attempt: We need to find a bijective function $f$ from $[0,1]$ to $[0,1)$ such that $f$ ...
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1answer
55 views

Question about the proof that the Hilbert Cube is compact.

Because of the fact that $(1)$ The topological space $[0,1]$ is a continuous image of the Cantor space $(G,T)$. There exists a mapping $\phi_n$ of $(G_n, T_n)$ onto $(I_n, T'_n)$ where, for each ...
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1answer
20 views

Topology .. cluster points

prove that if x is a cluster point of A unoin B then x is a cluster point of A or B I proved it by contrapositive but I want to prove it with direct proof
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Existence of sequence of polynomials such that $\lim_{n\to\infty} \int_0^1 |h(x) - p_n(x)|^2 dx = 0$

For a function $h:[0,1] \to \mathbb{R}$: $$h(x) = \begin{cases} 1~~\text{for}~~ x\in[0, \frac12] \\0 ~~\text{for}~~ x\in(\frac12, 1] \end{cases}$$ how could we prove the existence of sequence of ...
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problem in topology. looking for conditions under which given topology is discrete? [on hold]

Let $\tau$ be the topology on $\mathbb{R}$ for which the intervals $[a, b), -\infty < a< b < \infty$, form a base. Let $\sigma$ be a topology on $\mathbb{R}$ such that $\sigma \supseteq \tau$....
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Mandelbrot set and times tables

I recently saw a mathologer video on YouTube titled Times tables, Mandelbrot set and the heart of mathematics. It was about generating patterns using tables of numbers. I don't have any idea about it. ...
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2answers
43 views

Proof verificication and question of rigour: $A$, $B$, connected implies union is connected

Don't mark this as duplicate. The other question does not help me figure out how rigorous MY proof is. Problem: Let $A$ and $B$ be connected subsets of a metric space and let $A\cap\overline{B}\neq\...
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2answers
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Metric space where each continuous function has IVP is connected

The question: Let $X$ be a space such that every continuous function $f:X\rightarrow\mathbb{R} $ does have the following property: if $a<c<b$, $f(x) =a$, and $f(y) =b$, then there exists $z\in ...
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1answer
59 views

Embedding and homeomorphism

Suppose there exists an embedding from one topological space into another, and conversely. Is it always true that there is a homeomorphism between the two spaces?
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1answer
27 views

Pullback of a local homeomorphism is a local homeomorphism

Suppose we have pullback diagram of topological spaces: I want to prove: If $g:Y\to Z$ is a local homeomorphism (etale map), then $p_1:P\to X$ is a local homeomorphism. My idea: First of all $...
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1answer
23 views

Neighborhood of inclusion in space of Lipschitz maps is 1-1

Let $B \subset \mathbb{R}^n$ be the closed unit ball. Let $i(x) = x$ denote the inclusion map. Let $\|\cdot\|$ be any norm. Given $f:B\to \mathbb{R}^n$, define the sup norm $\|f\|_\infty:=\sup_{x \in ...
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2answers
31 views

Union of path connected pairwise not disjoint subsets

Problem Let $(X,d)$ be a metric space and let $\mathcal A$ be a family of path connected subsets of $X$ such that for every pair of sets $A,B \in \mathcal A$ there are $A_0,...,A_n \in \mathcal A$ ...
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0answers
19 views

Topological (Hierarchical) Ordering of the Data

I want to graph a data depicting the correlation and frequency. Mathematics-wise, it's not hard. It's matter of how to construct the algorithm for sorting and ultimate graphing (but more importantly ...
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1answer
51 views

Proof that a mapping onto $[0,1]$ is continuous.

Let each $(A_i,T_i) = (\{0,2\}, T_{discrete})$ and define $\phi : \prod (A_i, T_i) \rightarrow [0,1]$ with $\phi (<a_1, a_2, ...>) = \sum^{\infty}_{i=1} \frac{a_i}{2^{i+1}}$. In order to show ...
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1answer
63 views

When do minimal subcovers always exist - without choice?

In my answer to Doubt in the definition of a compact set, I sketched a proof of the following fact: Suppose $X$ is a topological space such that every open cover of $X$ has a minimal subcover. ...
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3answers
38 views

Show finite complement topology is, in fact, a topology

My attempt to prove the following is below: Let X be an infinite set. Show that $\mathscr{T}_1=\{U \subseteq X : U = \emptyset $ or $ X\setminus U $ is finite $ \}$ My book calls this set the "...
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4answers
89 views

Doubt in the definition of a compact set

It's said a set $A$ is compact if for every finite cover $U$ of $A$ there exists a subset of $U$ which also covers $A$, let's say $U_1$. Assuming $A$ is a compact set, we must be able to find a ...
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1answer
29 views

Mapping of open subsets of product spaces.

Let each $(A_i,T_i) = (\{0,2\}, T_{discrete})$ and define $\phi : \prod (A_i, T_i) \rightarrow [0,1]$ with $\phi (<a_1, a_2, ...>) = \sum^{\infty}_{i=1} \frac{a_i}{2^{i+1}}$. If we let $W = \{...
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2answers
64 views

Is my intuition of dense sets correct?

I am working with the usual definition of a dense set, which is Let $U$ be any non-empty open subset of $X$. A set $A$ is dense in $X$ iff $A \cap U \neq \emptyset$. My highly informal and ...
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2answers
59 views

Constructing an $L^2$ space on the unit ring $\mathcal{S^1}$

Revised Question: Starting with $L^2[0,2\pi]$, does the canonical map $$[0,2\pi)\ni\theta\mapsto e^{i\theta}\in\mathcal{S^1}$$(with functions going across in the obvious way) turn $L^2[\mathcal{S^1}]$...
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2answers
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Show that the open set in a $C_2,T_4$ space is a countable union of closed sets, without metrisation

Now I have a topological space $X$ that is $C_2$ and $T_4$, and $U$ is an open set in it, I want to show that $U$ can be expressed as $\cup_{i\in\Bbb Z_+} F_i$ where $F_i$ are closed sets, without the ...
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246 views

Is there a nonabelian topological group operation on the reals?

Inspired by A binary operation, closed over the reals, that is associative, but not commutative. That question asks for a noncommutative semigroup operation on $\Bbb R$, for which right projection is ...
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1answer
58 views

Nonempty closed sets on a connected space imply nonemptiness of intersection?

I am dealing with just real line to make things little easier for me. Suppose we have a set $X=[0,x],X'=[x,\infty)$. For the sake of argument, assume both are closed and nonempty. Claim: By the ...
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How to characterise a real object in the 4d euclidian space-time?

I'm not well versed in topology, but I want to give a (at least slightly) formal definition of a generalization of a real system. A real system can be an animal, a stone, or a technical system like a ...
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4answers
70 views

What does it mean to have a “different topology”?

On a space, I understand the notion of having different metrics on the same space. It is, in layman's terms, different ways of defining a distance but on the same space. But I often see the term "...
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Question about: Prove that $\phi : \prod_{i=1}^{\infty} (A_i,T_i)$ onto $[0,1]$ is continuous.

My question is: How does choosing $N$ sufficiently large effect this proof? Prove that $\phi : \prod_{i=1}^{\infty} (A_i,T_i)$ onto $[0,1]$ is continuous. Let each $(A_i,T_i) = (\{0,2\}, T_{...
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Colouring arbitrary regions, in a 2D plane populated with bicolored points

How may I efficiently colour arbitrary regions, according to the majority captured points, in a 2D plane populated with bicolored points distributed according to some unknown distributions. I could ...
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2answers
62 views

If $X$ is homeomorphic to $Y$ then is $X/\sim$ homeomorphic to $Y/\sim'$?/

Let $f:X\to Y$ be a homeomorphism between topological spaces. Suppose we have an equivalence relation $\sim$ defined on $X$. Define an equivalence $\sim'$ on $Y$ by $y_1\sim'y_2$ iff $x_1\sim x_2$ ...
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77 views

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
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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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0answers
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
28 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 $...
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1answer
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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
27 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
41 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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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
76 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 \...
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3answers
138 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
34 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
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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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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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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
48 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 $\mathbb{R} \to \mathbb{R}^{\omega}$ ? ...