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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0
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2answers
305 views

Reference for closed map lemma

I would like to have a reference (book, page) for the following version of the closed map lemma: If a continuous function between locally compact Hausdorff spaces is proper (i.e. preimages of compact ...
13
votes
1answer
168 views

If every five point subset of a metric space can be isometrically embedded in the plane then is it possible for the metric space also?

Let $X$ be a metric space with at least $5$ points such that any five point subset of $X$ can be isometrically embedded in $\mathbb R^2$ , then is it true that $X$ can also be isometrically embedded ...
2
votes
2answers
69 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 $f$ ...
3
votes
1answer
57 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 ...
2
votes
2answers
74 views

Introductory Topology Book Recommendation for Economics

Would you please share your 2 cent on book recommendation for introductory topology book to graduate student in Economics. Have exposure to the first half of the yearlong analysis course in the ...
2
votes
1answer
200 views

Uncountable compact sets in ordinal topology

The question I am interested in is the following: Let $\omega_1$ be the first ordinal with an uncountable number of predecessors. We consider $X=[0,\omega_1]$ supplied with order topology. Prove ...
1
vote
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?
2
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0answers
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: ...
0
votes
1answer
25 views

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.
0
votes
1answer
64 views

Is the closure of a geodesically convex set convex?

Is the closure of a geodesically convex set convex? If so, is there a simple proof for it? In $ \mathbb{R}^n $ there is a simple proof for it through convergent sequences. How should I apply it on ...
1
vote
1answer
18 views

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 ...
-2
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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
-2
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1answer
43 views

problem in topology. looking for conditions under which given topology is discrete? [closed]

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 $\sigma \supseteq \tau$....
0
votes
2answers
41 views

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 ...
6
votes
1answer
121 views

The set of w*-continuous operators is closed for the weak* topology?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
2
votes
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\...
0
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2answers
25 views

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 ...
3
votes
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 $L^2[\mathcal{S^1}]$...
0
votes
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 ...
1
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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 $A_0,...,A_n \in \mathcal A$ ...
1
vote
1answer
40 views

Interior of A-closure, isolated points?? [duplicate]

I have been working on this question. Let A be an open set. Does Int(A-closure)= A? Here is my answer (edited after reading the comments): Let A = (0,1), where A is a subset of [0,1]. Then, A-...
0
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1answer
28 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 $...
0
votes
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 ...
3
votes
1answer
66 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. ...
1
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4answers
94 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 ...
0
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3answers
40 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 "...
3
votes
1answer
92 views

Prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous, then it is continuous from the right

I'm trying to prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous (where the topology of $\mathbb{R}$ is $\emptyset$, $\mathbb{R}$, and all sets of the form $(-\infty, a)$), then it is continuous ...
3
votes
1answer
63 views

core-compact but not locally compact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
3
votes
2answers
80 views

Triangulation of torus - understanding why

Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider simplicial complexes. I do not understand why the triangles ...
0
votes
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 = \{...
0
votes
2answers
37 views

A covering space of a Hausdorff space is Hausdorff

Let $p:Y\to X$ be a covering space. If $X$ is Hausdorff, so is $Y$. Hello, I have a question to this task. I want to show that $Y$ is a Hausdorff space. Hence for $y_1, y_2\in Y$ with $y_1\neq y_2$ ...
3
votes
0answers
49 views

Extending the topology on a set to the group it generates

The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form $2^{n_2} \cdot 3^{n_3} \cdot 5^{n_5} \cdot ...$ The tuple ...
8
votes
2answers
249 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 ...
0
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2answers
22 views

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 ...
0
votes
1answer
61 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 ...
5
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1answer
66 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 ...
0
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0answers
28 views

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 ...
2
votes
5answers
64 views

Topology on $\mathbb N$ formed by taking the open sets to be $\emptyset, \mathbb{N}$ and $\{ 1, 2, 3, \ldots, n \} $ for each $n \in \mathbb{N}$

I am having some definition-wise problems. Problem: Prove that we get a topology for $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ by taking the open sets to be $\emptyset, \mathbb{N}$ and $\{ 1, 2, 3, \...
2
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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 "...
1
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0answers
31 views

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_{...
1
vote
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 $x_1\sim x_2$ ...
0
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0answers
9 views

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 ...
0
votes
1answer
55 views

Prove that there is no continuous surjection from $S^n$ to $\mathbb{R}_n$

Let $S^n = \{(x_1, . . . , x_{n+1}) \in \mathbb{R}^{n+1} \mid \sum_{k=1}^{n+1} x_k^2 = 1\}$. Prove that there is no continuous surjection $f : S^n \to \mathbb{R}^n$.
1
vote
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 $K$. ...
0
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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....
3
votes
0answers
79 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 ...
5
votes
3answers
145 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 ...
5
votes
3answers
331 views

Why are scattered sets G-delta?

I am looking for an elementary proof of the following - The problem appears in a real analysis text of A. Bruckner: A subset $S$ of $\mathbb{R}$ is called scattered if every non empty subset $X \...
0
votes
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 $f$ ...
0
votes
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 ...