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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2
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0answers
23 views

Any two maps to a cone space are homotopic.

I have to prove that any two continuous functions to a cone space are homotopic. Definition of cone space: If $Y$ is any topological space and $I=[0,1]$ is the closed unit interval in $\mathbb R$, ...
2
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0answers
18 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
2
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1answer
25 views

constructing a CW Complex

I am looking at an example of constructing a CW complex for a space X. The example i am looking at is that for The quotient of $S^2$ obtained by identifying north and south poles. The solution is as ...
0
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1answer
28 views

metric space: equivalence of several mertric.

I have two questions: Q1) Are all metric on a metric space are equivalent ? Q2) If not: Let $d_1,d_2$ two metric on $X$. If something has a property with a $d_1$ will it hold for $d_2$ too ? For ...
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2answers
37 views

Finding a choice for Epsilon for open/closed set proofs

I'm studying the proofs for open/close sets by using the following definition: I'm having problems to understand the proofs. The proofs sounds pretty straightforward: just choose a value for ...
-2
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1answer
14 views

an equivalent condition for compactness of a metric(topologic) space.

Let $X$ be a metric(or topological) space. 1) If every continuous function $f:X \rightarrow \Bbb R$ has a bounded image, then is $X$ a compact space? 2) If every continuous function $f:X \rightarrow ...
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2answers
19 views

A question involving continuity with respect to the product topology

Let H be a nonempty set, $\cdot$ a binary operation on H, $\Gamma$ a topology on H and $$\varphi : H \times H \to H, \;\; \varphi(x, y) = x y, \;\; \forall x, y \in H$$ continuous with respect to the ...
0
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0answers
19 views

Idempotent ideal in ring of continuous functions

Is there any equivalence conditions under which an ideal $I$ in ring of continuous functions be an idempotent ideal?
2
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2answers
41 views

Homeomorphism on the Hilbert space

We can consider two different topologies on the Hilbert space ; $l^{2}(\mathbb{N})$. One is the topology deduced from the norm \begin{equation*} \|f\|=\sqrt{\sum_{n=1}^{\infty} f(n)^{2}}, ...
6
votes
1answer
47 views

Does every continuous map between compact metrizable spaces lift to the Cantor set?

I'm interested in the universal properties of the Cantor set. It is well-know that the Cantor set $2^\mathbb{N}$ is "universal" in the category of metrizable compact spaces, in the sense that every ...
2
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2answers
121 views

$\epsilon$-dense subsets on $\mathbb R/\mathbb Z$.

Let $\langle M, d\rangle$ be a metric space. We say that $A \subset M$ is $\epsilon$-dense if every open ball of radius $\epsilon$ contains a point of $A$. Now let $T=\mathbb R/\mathbb Z$, the ...
0
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3answers
22 views

A countable Tychonoff space is normal?

I am trying to prove a countable tychonoff space must be normal but I cannot. Here is my work so far: We take two disjoint closed sets $F_1$ and $F_2$. Since $F_1$ is countable and $F_2$ is closed ...
3
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1answer
37 views

Explicit homeomorphism between open and closed rational intervals?

Sierpiński's theorem states that every countable metric space without isolated points is homeomorphic to $\mathbb{Q}$. (A proof can be found here and a discussion here). An immediate corollary is ...
2
votes
0answers
42 views

What's the most general geometry branch?

What is the most general geometry of curves and surfaces? For example, at curves, we define in differential geometry the tangent vector as the derivative of a regular curve, but visually many other ...
0
votes
1answer
29 views

$\mathbb{R}^{2}$ and $\mathbb{R} \times [0, +\infty]$ are homotopy equivalent, but not homeomorphic

So, let's consider $M=\mathbb{R}^{2}$ and $N= \mathbb{R} \times [0, +\infty]$ - two topological spaces. Since $\pi_{1}(M)=\pi_{1}(\mathbb{R}) \times \pi_{1} (\mathbb{R}) = \{0 \}$ (since $\mathbb{R}$ ...
0
votes
0answers
19 views

Does total order imply linearisation

Suppose $X$ is a totally ordered set. Does this mean that $X$ can always be linearised? I mean can $X$ be always written in a linear order like $\mathbb{R}$ ? I came across this question when I was ...
0
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2answers
37 views

Group homomorphism on unit circle

For $n\in \mathbb{Z}$, define the map $f_n:S^1\to S^1$ as $f_n(z)= z^n$, where the unit circle $S^1$ is observed as the subspace $\{z\in\mathbb{C}|\ |z|=1\}$. How would one compute the induced group ...
1
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2answers
24 views

Extreme value theorem, without Heine Borel.

I was wondering, if there are any mistakes, in this proof of the extreme value theorem: Theorem. Let $X$ be a compact set and $f:X\rightarrow\mathbb{R}$, s.t. $f$ is continuous. Then there exists ...
4
votes
1answer
50 views

The union of a sequence of closed sets with empty interiors has empty interior in a compact Hausdorff space?

This is problem 5 in section 27 of Munkres' TOPOLOGY, 2nd ed Let $X$ be a compact Hausdorff space; let $\{A_n\}_{n\in \mathbb{N}}$ be a countable collection of closed sets of $X$. If each set $A_n$ ...
1
vote
1answer
44 views

Does proper map $f$ take discrete sets to discrete sets?

Suppose $f:X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Are the following results true? $1$. The map $f$ takes discrete sets to discrete sets. $2$. If $f$ is ...
1
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1answer
45 views

Prove that the convergent sum of a real sequence is a metric

I want to show that $$ \varrho(\{a_n\},\{b_n\})=\left(\sum_{n=0}^\infty{(a_n-b_n)^2}\right)^{1/2} $$ is a metric, where $\{a_n\}_{n\in\Bbb N}\in \ell_2$, and $\ell_2$ is the set of all real sequences ...
4
votes
1answer
53 views

Questions about a topological proof of the FTA

I'm a high school student, curious about proofs of the Fundamental Theorem of Algebra. Specifically, I've been thinking about one of the topological proofs of the theorem, given in Courant's book, ...
0
votes
1answer
30 views

Non-empty intersection of specific sets

For any set Y (to begin with, it may be countable), given a collection of relations $$R = \{R_y \subseteq \{0,1\}^Y \mid y \in Y\},$$ having the finite intersection property and such that for ...
1
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1answer
24 views

Every nontrivial linear functional is open

Let $X$ be a normed linear space and let $f:X\to \mathbb K$ be a nontrivial linear functional. I want to prove that $f$ is open. I tried as follows: Let $E$ be an open set in $X$ and let $y\in f(E)$. ...
0
votes
1answer
41 views

does the closure of interior of a set equal to closure of this set?

Does the $\text{Cl}(\text{Int} A)=\text{Cl}(A)$? Here "Cl" denotes closure, "Int" denotes interior. That is not a duplicate of the question of "does the closure of interior of a set equal t the ...
2
votes
3answers
51 views

Does there exist a surjective continuous map $D^2 \to S^1$?

By considering the induced homomorphism on the fundamental groups, we know that there is no retract $D^2 \to S^1$. But is there any continuous surjection from $D^2$ to its boundary? It seems unlikely ...
2
votes
2answers
48 views

Topological proof that the interval $[a,b)\subset \mathbb{R}$ is not closed

I want to prove that the interval $[a,b)\subset \mathbb{R}$ is not closed using the definition that a set $A$ in a topological space $X$ is closed iff its complement $X-A$ is open. Here, the topology ...
3
votes
1answer
44 views

In $\mathbb Q_p$, proving every open ball is the disjoint union of more than one open ball

I'm reading the Foundations chapter of Gouvea's p-adic Numbers: An Introduction, and I'm trying to solve the following problem he poses to the reader: Take the $p-$adic absolute value on $\mathbb ...
3
votes
1answer
35 views

Remove one ring of Borromean rings in 3-sphere: linked or unlinked?

We know Borromean rings in a 3-sphere $S^3$ can be unlinked if we remove one of the three rings. Here let us consider a slight different procedure. If we remove the neighbored solid torus $B^2 \times ...
-1
votes
0answers
37 views

Homeomorphism from unit ball to unit sphere [on hold]

Consider the unit sphere in $\Bbb R^3$ given by $\{(x,y,z) \in \Bbb R^3|x^2+y^2+z^2 =1\}$. Let $p$ be a point in this unit sphere. Question: How can I construct an open set U within this unit sphere ...
-5
votes
2answers
40 views

proving f is continuous iff it takes limits to limits [on hold]

How to show that iff $x_i\to x$ implies $f(x_i)\to f(x)$ then f is continous? For metric Spaces. Continuit definition the standard epsilon delta
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0answers
19 views

Mollifiers and smooth path connetction

Generaly it is about smooth path connection. I have a two smooth paths f,g and a points x,y,z of open subset of $R^n$ such that f is smooth path from a to y and g is smooth path from y to z. I define ...
0
votes
1answer
18 views

Can a point $z$ which belongs to a closed set be a limit point of an open set which is disjoint from the closed set in topological space $X$?

Say $X$ be a topological space, and $U$ and $V$ are open and closed sets respectively. Furthermore, $U$ and $V$ are disjoint. Now there is a point $z \in V$. Is it possible for the point $z$ to be a ...
0
votes
1answer
38 views

Point-set topology

I am about to begin a self-study project in point-set topology. I am a final year undergraduate. I am looking for suitable resources, I have so far come across Munkres' textbook and would like to find ...
0
votes
1answer
36 views

Prob. 3 (b), Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How does the $K$-topology on $\mathbb{R}$ differ from the usual topology?

Let $$ K \colon= \left\{\ \frac{1}{n} \ \colon \ n \in \mathbb{N} \ \right\},$$ and let the $K$-topology on $\mathbb{R}$ be the one having as basis all open intervals $(a,b)$ and all sets of the form ...
0
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1answer
33 views

Showing the attractor of an IFS is either connected or totally disconnected

I came across this execise in a problem set about Iterated Function System (IFS) and fractals: "Show that the attractor of an IFS of the form $\{\mathbb{R};~ax+b, cx+d\}$ where $a,b,c,d \in ...
0
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0answers
26 views

The Coproduct of two spaces is the same as the disjoint union and is homeomorphic to the union when the spaces are disjoint

Let $X$ and $Y$ be two topological spaces. Consider the set $$X \cup Y = \{ x\; |\; x \in X \text{ or } x \in Y\}$$ In all the following I suppose that $X$ and $X$ are disjoint. I want to ...
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vote
3answers
73 views

What are some simple examples illustrating the definition of “cover”

In my class the word "cover" is used very informally such as this set covers another set (this is for a class in PDE not topology by the way). Can someone provide a trivial example of cover to get ...
4
votes
9answers
463 views

What is the mathematical distinction between closed and open sets?

If you wanted me to spell out the difference between closed and open sets, the best I could do is to draw you a circle one with dotted circumference the other with continuous circumference. Or I would ...
3
votes
0answers
57 views

Hatcher 2.1.10…

Hatcher asked a question in chapter 2 (a) Show the quotient space of a finite collection of disjoint 2-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic with ...
0
votes
2answers
34 views

A base generates an unique topology?

I was confused by this. Let $X$ be {$a,b,c$}, Let $\mathcal{B}$ be {{$a$},{$b$},{$c$}}. Let $ \mathcal{T}$ be {$X, \emptyset$, {$a$}, {$b$}, {$a,b$}}. Let $ \mathcal{T'}$ be {$X, \emptyset$, ...
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0answers
16 views

Disjoint Union of Completely Regular Spaces

I am trying a new approach to an already-solved problem, but I need help to see if I'm on point. Munkres Chapter 53, question 6 [abridged] asks, given a covering map $p: E \to B$: Show that "if $B$ ...
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votes
2answers
62 views

Decomposing 2-sphere into two homeomorphic subspaces [on hold]

Can a 2-dimensional sphere be decomposed into two disjoint homeomorphic subspaces? If yes, can these subspaces be non-discrete / connected / have some other good properties?
0
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0answers
23 views

Construction of a Radon measure from a certain family of compact subsets

Let $X$ be a locally compact Hausdroff space. Let $\Gamma$ be a family of compact subsets of $X$ with the following properties. 1) $\emptyset \in \Gamma$. 2) $K\cup L \in \Gamma$ whenever $K ...
1
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1answer
41 views

simplicial homology definition

I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105: Hatcher ...
1
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1answer
27 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
0
votes
1answer
24 views

Boundary preserving map

Let $K\subseteq\mathbb{R}^2$ be a compact set. Is it true that for a continuous map $p:K\to\mathbb{R}^2$ we have: $p(\partial K)=\partial p(K)$? Are there any generalizations? P.S. Note that ...
0
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0answers
26 views

Pro-completion of finite algebras as Stone algebras

Recall that a profinite algebra (e.g. group, monoid, or whatsoever) is a cofiltered/inverse limit of finite algebra. In Johnstone's Stone space, he showed that finite discrete algebras are finitely ...
2
votes
1answer
29 views

Show that if $(X,d)$ is compact then, every open covering of $X$ has a Lebesgue number.

Let $(U_i)_{i \in I}$ be an open cover of a metric space $(X,d)$, a number $\epsilon >0$ is called a Lebesgue number of $(U_i)_{i \in I}$ if for all $x \in X$ exist $j \in I$ such that ...
3
votes
2answers
46 views

An infinite dimensional normed linear space is the union of two disjoint convex sets

Let $X$ be an infinite dimensional normed linear space. I want to show that there exist two disjoint convex sets $C_1$ and $C_2$ such that $X=C_1\cup C_2$ and both $C_1$ and $C_2$ are dense in $X$. I ...