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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13 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
0
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1answer
28 views

If I have a function that's continuous and it's limits at $\pm \infty$ are $\pm \infty$ is it surjective?

I was trying out some problems where I needed to prove that a function was surjective, and I thought I could do this, is this true? Intuitively, it seems so. If I have a function that's continuous ...
3
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4answers
49 views

For any closed subset of $\mathbb{R}$ there is a sequence in $\mathbb{R}$ whose sequential limits is equal to the that subset

Question: Let $A$ be a closed subset in $\mathbb{R}$. Prove that there exists a sequence $x_n$ in $\mathbb{R}$ whose set of subsequential limits is exactly equal to $A$. My approach: I think this ...
5
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2answers
36 views

On any continuous map $f:S^1 \to \mathbb R$

Let $f:S^1 \to \mathbb R$ be any continuous map , where $S^1$ is the unit circle in the plane . Let $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ ; then how to prove $A$ is uncountable , or ...
1
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1answer
29 views

How to know which notion of convergence to use when proving density of a subspace

My question might be a little vague, but is there a way to know which type of convergence (i.e pointwise, uniform) to use when proving that a subspace is dense in a certain space. For example if we ...
1
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0answers
36 views

Integrating the Hopf invariant for $\pi:S^3\to S^2$

I've been working on the last part of problem 9., chapter 9 in Nakahara's Geometry, Topology and Physics all day, with no success, and am in need of some assistance. We are asked to compute the Hopf ...
4
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4answers
83 views

On extended real line, is $(-\infty,+\infty)$ still a closed set?

On real line $(-\infty,+\infty)$ is open as well as closed. On extended real line $[-\infty,+\infty]$, is $(-\infty,+\infty)$ still a closed set? Thank you.
0
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0answers
11 views

Topology induced by quasi-pseudo-metrics

So I know that uniform spaces can be described using a collection of metrics which satisfy 2 properties. For a topological space $(X,\mathcal{T})$ you can define a collection of pseudo-quasi metrics ...
2
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3answers
32 views

If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0$ always? [duplicate]

If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0?$ My attempt: Suppose $dist~(E,F)=0.$ Then $\exists~e \in E,f \in F$ such that $~\forall ...
0
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0answers
59 views

Engelking or Munkres for General Topology? [on hold]

I am a last-year Bachelors student and when I finish my Bachelors, I will have one year free time just before I start Masters in Pure Mathematics. I like to be better in General Topology before ...
1
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1answer
20 views

Every countably infinite subset of a countably compact space has an $\omega$-cluster point

First the definitions: The point $p$ is an $\omega$-cluster point for a subset $A$ of a topological space $X$ if every neighbourhood of $p$ contains infinitely many points of $A$. A space is ...
0
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0answers
70 views

If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold? how can String theory state that there are only $11$ dimensions?

I am reading some basic context books about topology (i.e. The Poincaré Conjecture, by Donal O'Shea between others) and following this open Topology and Geometry video lectures of the brilliant ...
4
votes
1answer
45 views

Is an open subset of a compact surface with connected boundary completely determined by its fundamental group?

Is an open, connected subset of a compact surface with connected boundary determined (up to homeomorphism) by its fundamental group? If we weaken the hypotheses, I can see how this can fail: A ...
2
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2answers
45 views

Whether the set $A$ is connected

Let $f: \mathbb R \rightarrow \mathbb R$ be continuous function and $A\subset \mathbb R$ be defined by $$ A=\{y\in R : y= \lim_{n\rightarrow \infty} f(x_{n}) \text{ where } x_n \text{ diverges ...
2
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1answer
34 views

Name for continuous maps satisfying $\operatorname{cl}(f^{-1}f(U))= \operatorname{cl}(U)$

I have recently come across particularly kind of continuous maps $f \colon X \to Y$ between topological spaces with the property that $$ \operatorname{cl}(f^{-1}f(U))= \operatorname{cl}(U), $$ for ...
5
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1answer
32 views

Which is the domain set?

Let $X \subseteq \mathbb{R}$ and $f,g : X \rightarrow X $ be continuous functions such that $f(X) \cap g(X) = \emptyset$ and $f(X) \cup g(X) = X$. Then which of the following cannot ...
0
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0answers
11 views

Are trace of z-filter in dense z-embedded subset z-filter?

I found this article about z-filter, referring to Lemma 3 my question is: without the "every member of which meet Y" hypothesis and adding that Y has to be dense in X is it still true? EDIT: forgot ...
2
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1answer
33 views

Homotopy of certain maps induced homotopies

Let $\psi$ be a homeomorphism and $\gamma :[0,1] \rightarrow \mathbb{R}^2$ a path. Now assumme additionally that $(\psi \circ \gamma)(t) \neq \gamma(t)$ everywhere. Then we can look at the map ...
4
votes
1answer
95 views

Image of a continuous function

Let $f :\mathbb R \rightarrow\mathbb R$ be continuous function . Then which cannot be the image of $(0,1]$ ? A. $\{0\}$ B. $(0,1)$ C. $[0,1)$ D. $[0,1]$ Now A. is ...
2
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0answers
43 views

General topology exercise (equivalent condition for simple connection)

Let $X$ be a pathwise-connected topological space. Prove that $X$ is simply connected iff every continuous $f:S^1\to X$ can be extended to a continuous function $g:D^2\to X$. How can I use the fact ...
1
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0answers
27 views

A mistake in a proof of consistent choice?

Given a set of sets ${\cal A} = \{S_i\mid i\in {\cal B}\}$ and a binary relation $Con$ on $\bigcup {\cal A}$, a $Con$-choice on $\{S_i\mid {i\in F}\}, F\subseteq {\cal B},$ is a function $\epsilon\in ...
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0answers
22 views

Prove the result on connected sets in complex analysis. [on hold]

If $B = S \cup \{$some or all of its limit points$\}$, then $B$ is connected.
3
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1answer
48 views

Insight about compact groups

I'm quite familiar with the general notion of compactness in math but I have some troubles with its extension to group theory. I'm not talking about definitions or theorems: I would like to have some ...
2
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1answer
37 views

The topology of $\mathbb{Z}_p$

I don't know much about topology, but anyway... Assuming $\displaystyle\prod{A_n} =\prod_{n\geq 1}{A_n}$, why is $\mathbb{Z}_p$ closed in a product of compact spaces? Googling I found Tychonoff's ...
3
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0answers
38 views

The Weak topology on an infinite-dimensional space is not metrizable

Let $X $ be an infinite-dimensional normed space I want to prove that weak topology on $X$ is not metrizable, this is my solution Assume that there is a metric $d$ on $X$ such that induced weak ...
2
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0answers
68 views

Is there a connected linearly ordered space of size $\leq 2^\omega$ that looks nowhere like the reals?

I want a connected linearly ordered space with cardinality $\leq 2^\omega$, such that no piece of it looks like an interval of reals. Is this possible? Edit: I'd rather not assume extra axioms, but ...
4
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0answers
35 views

Spaces whose all their metrizations are complete [duplicate]

Which metrizable topological spaces $(X,\tau)$ posses the following property: Every compatible metric (i.e one which induces the same topology $\tau$) is complete. Compact metrizable spaces satisfy ...
0
votes
3answers
53 views

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|$

$\mathbb{N}$- a complete metric space with $d(x,y)=|x-y|.$ This seems quite intuitively correct, but I do not know how to prove this formally, does anyone know how they would go about this?
1
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2answers
47 views

Show $f: S^1 - {N} \to \mathbb{R} $ $f(x_1,x_2) = \frac{x_1}{1-x_2}$ is Homeomorphism

$S^1$ is a unit circle and $N := \{ (0,1) \in S^1\}$. The question hints that the for any $(x_1,x_2) \in S^1- {N}$, line joining $N$ and $( x_ 1 , x_ 2 )$ meets the $x$ -axis at ($f ( x_ 1 ;x_ 2 ) , 0 ...
3
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3answers
115 views

Show that $\mathbb{R}^2/{\sim}$ is homeomorphic to the sphere $S^2$.

$\mathbb{R}^2/{\sim}$ is the smallest equivalence relation such that $P\sim Q$ for all $P,Q$ with $\|P\|_2,\|Q\|_2 \geq 1$. Show that $\mathbb{R}^2/{\sim}$ is homeomorphic to the sphere $S^2$. ...
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0answers
27 views

commutativity taking the complement and taking fibers

Let $\mathcal M \rightarrow S$ be a projective irreducible scheme over the spectrum of a DVR and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and ...
2
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0answers
51 views

How to draw the knot 2, -32, 41?

Hej, I have the following exercise: Draw the tangle 2, -32, 41 and the corresponding knots obtained by connecting the NW string to the NE string and the SW string to the SE string. (From C. C. ...
1
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0answers
30 views

How do I prove that this product space is normal?

Let $A$ be a compact subspace of $\mathbb{R}^2\setminus\{0\}$. Let $C$ be a connected component of $\mathbb{R}^2\setminus A$. Define $D=C\cup A$. How do I prove that $D\times [0,1]$ is normal? I ...
3
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1answer
26 views

Are Homogenous countable complete metric spaces always discrete?

Let $M$ be a countable complete metric space such that the group of isometries of $M$, $Iso(M)$ acts transitively on the points in $M$. Does it follow that the topology induced by the metric is the ...
0
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2answers
92 views

Why is every point in an open interval $(a,b)$ not a limit point?

If I have an open interval $\Bbb R\supset A=(a,b)$ then I can pick any $x:a<x<b$ and make a ball with center $x$ which contains a point inside the interval. However, this article from proofwiki ...
2
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1answer
67 views

Dedekind Construction Of Real Numbers

If we define Dedekind-real numbers as Dedekind cuts, i.e. $\sqrt 2 = \{\text{rationals less than }\sqrt2\} \cup \{\text{rationals more than } \sqrt2\}$, can we define addition and multiplication of ...
4
votes
1answer
61 views

Can we parameterize a topological space?

It has been few months since I started doing topology . There was this idea which struck me a few days ago . For example the parametrization of a line is $$x=qv+a,$$ where $t$ is the parameter. ...
1
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0answers
26 views

Is $\mathbb R^N$ an $C$-distinguished topological space?

I am reading a paper which has some complicated construction on a Hausdorff topological space called $C$-distinguished topological space. The paper says that a $C$-distinguished topological space $X$ ...
6
votes
4answers
104 views

Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
3
votes
1answer
57 views

Is Heisenberg group Euclidean?

I'm reading an article speaking about Heisenberg group $\mathbb H^n$ and some of its properties. Now, I have some questions to ask, hoping to be clear enought. Reading the introduction I've ...
2
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0answers
48 views

Topologies on the collection of $\sigma$-algebras

Let $X$ be a non-empty set and let $\mathfrak S$ be the collection of all $\sigma$-algebras on $X$. That is, a typical element $\mathscr S\in\mathfrak S$ is a $\sigma$-algebra on $X$. For example, ...
5
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1answer
75 views

How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...
13
votes
3answers
381 views

Is a space compact iff it is closed as a subspace of any other space?

I am trying to come up with an alternate definition of a compact topological space that coincides with the usual one. Sorry if my topology is a little rusty. My proposed alternative definition is ...
2
votes
1answer
36 views

Specific question on $l^p$ spaces and its dual in weak * topology

I am covering now Lp spaces in my summer real analysis course and this problem from Folland related to the dual of Lp stumped me hard, it is problem 19 chapter 6 reads as follows: We define $ ...
3
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0answers
40 views

Proving that $S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$ is open in $C^2[0,1]$ with a specific metric

I am trying to prove that $$S^c=\left\{f(x)\in C^2[0,1]\;\Big\vert\; \int_{0}^{1}f(x)dx > 3\right\}$$ is open in $C^2[0,1]$ with the metric $d$ given by $$ d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ ...
0
votes
1answer
27 views

Feedback on my solution “Determine a set compact or not”

Let $X :=\{(x_1,0,x_2) \in \mathbb{R^3}, x_1, x_2 \in \mathbb{R}\}$ $\mathcal{T}$ be subsapce topology coming from standard topology on $\mathbb{R^3}$. My answer is that it's compact. Reason: Define ...
2
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1answer
40 views

Completeness of ${C^2[0,1]}$ with under a specific metric

Prove that ${C^2[0,1]} $ (set of two times differentiable functions)is complete with metric: $$d(f,g)=\sup_{x \in [0,1]}|f(x)-g(x)|+ \sup_{x \in [0,1]}|f'(x)-g'(x)| + \sup_{x \in ...
13
votes
5answers
615 views

Why formulate continuity in terms of pre-images instead of image?

I wanted to discuss my intuition of why we formulate the concept of continuity in terms of pre-image of open set is open instead of images for example if we consider $f(x) = c$ where $c$ is some ...
4
votes
1answer
33 views

Topologically distinguishing Mobius Strips based on the number of half-twists

We can distinguish between a (closed) Mobius strip and 'regular' (untwisted) strip by examining the set of points which have no neighborhood homeomorphic to a disk (intuitively, the 'boundary' of the ...
5
votes
0answers
48 views

What are the universally effective epimorphisms of topological spaces?

An effective epimorphism in a category is a morphism that is the coequaliser of its kernel pair, and a universally effective epimorphism is a morphism $f : X \to Y$ such that, for every pullback ...