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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20 views

Degree 3 map from the torus to the sphere.

Construct a degree 3 map from $T$ to $S^2$ where $T$ is the torus? I can find a degree 0 and degree 1 map by the following proof: Embed $T$ in $\mathbb{R}^3$ in the usual way. Consider any point ...
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1answer
33 views

Composition of homeomorphisms $[0,1] \to [0,1]$

Let $f:[0,1] \to [0,1]$ be a homeomorphism with $f(0)=0$ and $f(1)=1$. If $f$ is not the identity map, is it true that $f^n \neq f$ for all integers $n>1$? Edit: By $f^n$ I mean $f$ iterated $n$ ...
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2answers
44 views

$(X \times X) /{\sim'}\cong (X/{\sim}) \times (X/{\sim})$

The full description of this problem is: Let $X$ be a topological space. Let $\sim'$ be the equivalence relation on $X\times X$ defined by $(x,y)\sim'(x',y')$ iff $x \sim x'$ and $y \sim y'$ ...
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2answers
26 views

Covering map between two path connected sets

First off, I see a lot of variations of this problem cropping up on practice qualifiers, and I'm trying to regain my knowledge of topology. Let $p: X \to Y$ be a covering map where $X$ and $Y$ are ...
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1answer
35 views

Infinitely countable subset of $\mathbb{R}^2$ is connected.

Let $A\subset \mathbb{R}^2$ be an infinite countable subspace. Can $A$ be connected? I have seen plenty of proofs that show that $\mathbb{R}^2\setminus A$ is connected and I understand those. ...
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2answers
58 views

simple proof for principle of pigeons

I must prove the principle of pigeons but the proofs I find in the internet are too complex. Here's what I can use: Definition $$I_n = \{p\in \mathbb{N}; p\le n\}$$ The principle of the pigeons ...
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0answers
16 views

Understanding exponential function [on hold]

It´s me again. Consider $\phi:\left(-1,1\right)\longrightarrow\Bbb S$ \ $\lbrace-1\rbrace$ where $\Bbb S=\lbrace z\in \Bbb C: \vert\vert z\vert\vert=1\rbrace$ $\phi\left(t\right)=e^{i\pi t}$ My ...
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1answer
27 views

Let $X$ be a compact space such that all points are isolated. Then $X$ is finite.

Let $X$ be a compact space such that all points are isolated. Then $X$ is finite. Some help, thanks.
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1answer
21 views

Convergence of subnets

Suppose that exists a subnet of a net $( x_{\mu})$ in $X$ that converge to $x \in X$, then $x$ is a cluster point of $( x_{\mu})$. My attempts were: suppose $x$ is not a cluster point of $( ...
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1answer
18 views

Assume that for all $x\in A$ there exists a neighborhood $S$ of $x$ such that $f$ is constant in $S\cap A$. Then $f$ is constant in $A$

Let $f:A\subset X\to Y$ be a continouos in the connected set $A$. Assume that for all $x\in A$ there exists a neighborhood $S$ of $x$ such that $f$ is constant in $S\cap A$. Then $f$ is constant in ...
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2answers
23 views

Example of a bounded space which is not totally bounded

I was trying to find an example of a bounded metric space which is not totally bounded. The only example I could come up whith was the natural numbers with the discrete metric. However, like any ...
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1answer
22 views

Let $\mathcal S$ be the collection of all straight lines in the plane $\mathbb R^2$. If $\mathcal S$ is a subbasis for a topology …

Let $\mathcal S$ be the collection of all straight lines in the plane $\mathbb R^2$. If $\mathcal S$ is a subbasis for a topology $\mathcal T$ on the set $\mathbb R^2$, what is the topology? I ...
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0answers
40 views

I want to find questions or problems to do math research. [on hold]

This is a soft question, but I am not sure of a better forum. I am looking for a professor who can help me find a question or problem that I can independently research and try to solve in order to ...
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3answers
42 views

Complex Analysis - what makes a simple connected set?

Having difficulty finding the differences between a connected set and a simply connected set and a region. Would be good if someone could inform me and also give an example. Thanks Tom
2
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1answer
23 views

One-sided limit in topology

Can we define One-sided limit in topology ? I think our space must be order set (Partially ordered set) . is it true?
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1answer
21 views

Existence of an “open to closed” and “closed to open” function

Somebody knows if there exists a function $f:\mathbb{R}\to\mathbb{R}$, with it's usual topology, such that the image (or preimage) of every open set is closed and the image (or preimage) of every ...
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4answers
41 views

Show that $M$ is open in $\mathbb R^{n+1}$.

Show that $$M= \{ (x_1,x_2, \dots,x_n,x_{n+1})\in\mathbb R^{n+1} : x_1^2 + x_2^2 + \dots+x_{n+1}^2 <1 \}$$ is open in $\mathbb R^{n+1}$. I considered trying to show this using the easiest ...
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1answer
28 views

Under which additional hypothesis are open maps locally injective

Recollection of basic definitions: We recall the basic definitions that a continous map of topological spaces $f : X \to Y$ is open if $f(U)$ is an open subset of $Y$ whenever $U$ is an open subset ...
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1answer
39 views

Do non-second-countable spaces have “small” non-second-countable subspaces?

If $X$ is any space which is not second-countable, can one find a subspace $Y \subseteq X$ with $|Y| \leq \aleph_1$ which is also not second-countable? (Recall that a topological space $X$ is ...
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0answers
23 views

Where $X$ is a normed space, show $U_1=\{x \in X:||x|| <1 \}$ is an open set in $X$

Let $X$ be a normed space, show $$U_1=\{x \in X\mid \|x\| <1 \}$$ is an open set in $X$. This is the proof: Let $x \in U_1$, then $\|x\|<1$. Let $e=1-\|x\|$ so $e >0$. If $y \in X$ ...
7
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0answers
55 views

A List of Standard or “Cliche” Homeomorphisms [duplicate]

Learning topology has been hard. I just cannot see how some people can come up with complex functions that link one space to another, in a homeomorphic sense. The explanations are always "Well if you ...
2
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1answer
42 views

Help finding paper from the 1920's

I have not been able to find a copy of this paper anywhere! B. Knaster еt C. Kuratowski: Sur quеlquеs propriétés topologiquеs dеs fonctions dеrivéеs. Rеnd. dеl Сirc. Math. di Palеrmo, 59 (1925), ...
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1answer
51 views

Why this set is dense in $C_0(\mathbb{R})$

Let $C_0=\{f~|~ f:\mathbb{R}\to\mathbb{R},f~is~continous,\lim\limits_{\vert x\vert \to\infty}f(x)=0\}$ $A=\{f~|~f(x)=p(x)e^{-x^2},p(x)~is~polynomials\}$ Why $A$ is dense in $C_0$. The topology ...
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2answers
13 views

Accumulation point implies strict limit point?

Given a topological space $(X, \mathcal{T})$, and a subset $A$ of $X$, define $p$ to be a strict limit point of $A$ if there exists a sequence $(x_n)_n$ in $A \setminus \{p\}$ such that $x_n ...
4
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1answer
69 views

True or False: there is a space $X$ such that $S^1$ is homeomorphic to $X\times X$

I had an exam this morning, one of the questions asked about the truth of the statement There is a space $X$ such that $S^1$ is homeomorphic to $X\times X$. I said that this was false and this ...
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1answer
16 views

Is every set comeagre in its closure?

Let $A$ be a set in a topological space $X$. We know that $A$ is dense in its closure $\bar{A}$. This implies that $\bar{A} \setminus A$ is nowhere dense in $\bar{A}$ (using a characterisation of ...
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1answer
27 views

How to show that a set $D\subset R^2$ is not connected using a continuous function $F:R^2\mapsto R$

We have $$F:R^{2}\rightarrow R , F\left( x,y\right) =x^{3}+2x^{2}y+y^{3}$$ and for $D\subset R^{2}$ we know $$0\in F\left( D\right),4\in F\left( D\right) \text {but } 2\not\in F\left( ...
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1answer
22 views

Group action on coset space is continuous

I found this exercise in various places, but I could not find the answer anywhere. As I am quite new to topology, I would appreciate any help. Let $G$ a topological group and $H$ a subgroup. Let the ...
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1answer
50 views

finding sup and inf of $\{\frac{n+1}{n}, n\in \mathbb{N}\}$

Please just don't present a proof, see my reasoning below I need to find the sup and inf of this set: $$A = \{\frac{n+1}{n}, n\in \mathbb{N}\} = \{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, ...
0
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1answer
47 views

Unit sphere without a point is contractible

Let $a$ be a point on the unit sphere $S=\{(x,y,z)|x^2+y^2+z^2=1\}$. How do I show that $S\backslash\{a\}$ is contractible? How do I show that a non-surjective loop $\phi\in P(S,s)$ with ...
0
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1answer
32 views

Unique homomorphisms between fundamental groups of topological spaces

Let $u:A\rightarrow B$ be a continuous map of topological spaces, $a\in A$, $b=u(a)$. How do I prove that there exists a unique group homomorphism $$u':\pi_1(A,a)\rightarrow\pi_1(B,b)$$ ...
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0answers
17 views

Connected components of the complement of a geodesic

I came across the book "Knots, Molecules, and the Universe: An Introduction to Topology" by E. Flapan which is quite nice. In the first chapters it is discussed how one can distinguish the sphere ...
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1answer
39 views

Why is the set of all half-lines in the real numbers not a topological space?

In a basic topology text by Anatole Katok, the following example of a set that is not a topological space is given: If in the set of real numbers ${\mathbb R}$ we declare open (besides the empty set ...
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1answer
17 views

Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
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2answers
38 views

Two geodesics cannot form a simple region

Suppose S is an orientable surface with nonpositive Gaussian curvature. How can I prove that two geodesics that start from the same point $p\in S$ cannot meet again at another point $q\in S$ such that ...
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1answer
17 views

Is the intersection of 2 annuli in a non-archimedean space an annulus?

Let A be a ball in a non-archimedean topological space, $A = \{x: |a-x| < r \}$ and let A contain two Balls that are disjunct from each other: $B = \{x: |a-x| < s \}$ and $C = \{x: |b-x| < t ...
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0answers
21 views

Cellular homology for 3-sphere and $L^3$ lens space

When trying to follow the professor's computation of the cellular homology for the lens space $L^3(4)=S^3/\mathbb{Z}_4$ I became aware I had some trouble understanding the definition of the (cellular) ...
2
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1answer
31 views

Is the unit circle “stretchy” with respect to its norm?

Suppose we have a collection of metric spaces on $\mathbb{R}^n$, each of which has a different p-norm, $1\leq p \leq \infty$. ($p=2$ is Euclidean distance, $p=1$ is taxicab distance, etc.) Then, ...
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5answers
229 views

Complement of rationals has empty interior

This question refers to How to prove closure of $\mathbb{Q}$ is $\mathbb{R}$ I want to prove that the closure of $\mathbb{Q}$ is $\mathbb{R}$. I am trying to understand the accepted answer, but when ...
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1answer
40 views

How can one prove that a real function is closed? [on hold]

I am defining a closed function to be one that takes closed sets to closed sets. Given a function, domain and codomain, you could prove that it is not closed by simply providing a counter example ...
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2answers
39 views

Preimage of sets, complement of sets, continuity of functions

I just got some simple questions in real analysis regarding preimage and complement of sets and continuity. Suppose $f:X\to Y$, then does $f^{-1} (Y\setminus F)=f^{-1} (Y)\setminus f^{-1} ...
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1answer
15 views

Boundaries of Sets U and V

Let $(X,T)$ be a topological space and let $U$ and $V$ be subsets of $X$. a.) Then $Bd(U) − Bd(V) ⊆ Bd(U − V)$. b.) Then $Bd(U − V) ⊆ Bd(U) − Bd(V)$. I have come up with a pretty clear ...
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0answers
19 views

A circle with a line bisecting it is homotopic to the wedge sum of two circles

Let $C$ be the circle centered at $0$ with radius $2$. $L$ is the segment connecting $(0,2)$ and $(0,-2)$. Prove that $F = C \cup L$ is homotopic to the wedge sum of two circles. Intuitively, I ...
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0answers
43 views

Does this function have a dense graph?

Let $\mathbb Q =\{q_n:n\in\mathbb N\}$ be an enumeration of the rationals. Let $f(x)=\mid\sin(1/x)\mid$ if $x\neq 0$ and $f(0)=0$. Let $g(x)=\sum_{n=1} ^\infty \frac{f(x-q_n)}{2^n}$. Question: ...
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2answers
49 views

Fundamental group of a wedge sum, in general (e.g. when van Kampen does not apply)

What is the fundamental group of a wedge sum in general? e.g. including the times when van Kampen cannot help us. The Wikipedia article on wedge sums mentions that Van Kampen's theorem gives ...
2
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1answer
25 views

Is the surface of a torus 2-dimensional?

Unless I'm very mistaken, the surface of a torus is 2-dimensional, as is the surface of a sphere. The reason being that being on the surface you can only move in 2 dimensions, up or down is not well ...
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5answers
967 views

Is a ball always connected in a connected metric space?

If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?
2
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1answer
32 views

Graph of a $G_\delta$-function

Let $f:\mathbb R \to \mathbb R$ be a function. It is well known that if $f$ is continuous ($f^{-1} [A]$ is closed whenever $A$ is closed), then its graph is closed in $\mathbb R ^2$. Here is an ...
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2answers
63 views

Does there exist a Vector that can't be written as a Tuple of Scalars?

The most abstract/general definition of a vector The most general definition of a vector is as an element of a vector space. Given a vector $u$, we can always say that there exists a vector space $V$ ...
0
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1answer
24 views

Determine If This Is A Topology on U

Let $f:U \to V$ be a function and supposed that $T$ is a topology on $V$. Then {$f^{-1}(S):S \in T$} is a topology on $U$. I understand that I need to prove this or provide a counterexample using ...