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; ...

learn more… | top users | synonyms (3)

0
votes
1answer
9 views

Cardinal Function-Tightness

If $X$ is a space of uncountable many copies of the space $\mathbb{R}$ with the discrete topology. Then can we conclude that tightness of $X$ is an uncountable cardinal.
1
vote
1answer
25 views

Every open set S in $R^n$ can be expressed in one and only one way as a countable disjoint union of open connected sets.

In the following proof in my book their is something I don't understand I will present the proof then present the statement which I don't quite agree with. I want to also ask about some stuff I don't ...
2
votes
0answers
30 views

Proof that the closed interval in $\mathbb{R}$ is connected

Let $C$ be an open and closed subset of $[a,b]$. WLOG, assume $a \in C$. Set $A = \{x \in [a,b]: [a,x] \subseteq C\}$. Since $a \in A$, sup$A$ exists. Let $\epsilon > 0$. Then, (from real ...
0
votes
1answer
11 views

Is the hypercone contractible?

I know the cone CX is contractible but was wondering if the 3 dimensional cone in 4-space was also contractible.
5
votes
2answers
35 views

How to prove the not-so-long rays are homeomorphic to the reals?

The long ray (half of the long line) is an interesting topological space. It is defined as the order topology on $\omega_1$$\times [0, 1)$ with lexicographic order. Basically, it is an uncountable ...
0
votes
1answer
37 views

How can i write klein bottle as an adjunction space?

I want to find the homology groups of the klein bottle by Mayer-Vietoris. For this I want to describe the klein bottle as an adjunction space. I think it can written as a pushout $ S^1\cup_f D^2$ but ...
0
votes
0answers
12 views

homology group of adjunction space

I start to study homology theory and i want to understand homology groups of adjunction space In this picture i can't see $V$ deformation retracts to $X$ neither intuitively nor explicitly help ...
1
vote
0answers
14 views

Proof that the extreme points of a compact convex is not empty

The Krein–Milman theorem states that if $S$ is convex and compact in a locally convex space, then $S$ is the closed convex hull of its extreme points. In particular, such a set has extreme points. Is ...
0
votes
1answer
28 views

Showing that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$ [on hold]

I want to show that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$, however I am not to sure how to go about doing so. We have, $$d_\infty(x,y)=\sup_i|x_i-y_i|$$ and $\mathscr ...
0
votes
1answer
25 views

Does this metric induce the topology on the product space?

Consider $\left\{1,2,3\right\}$ with the discrete topology on it. Moreover, consider $Z=\left\{0,1,2\right\}^{\mathbb{Z}}$ with the associated product topology. The cylindersets $$ ...
3
votes
3answers
137 views

Can a space $X$ be homeomorphic to its twofold product with itself, $X \times X$?

Let $X$ be a topological space of infinite cardinality. Is it possible for any $X$ to be homeomorphic to $X\times X$ $?$ For example, $R$ is not homeomorphic to $R^{2}$, and ...
0
votes
2answers
27 views

Metric spaces, manipulating the absolute value function.

I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that, $$\sum_{i=0}^\infty x_i^2 \lt\infty$$ where $x_i$ is the $i$-th term in the infinite ...
0
votes
0answers
39 views

Is $\mathbb{Z}$ equipped with the discrete topology in this excerpt from Munkres' book?

I'm reading Munkres' Topology, p.385: Here, he is constructing a covering space $E$ of $X$, where $U,V$ are open subsets of $X$ such that $U\cup V=X$. I think he meant $\mathbb{Z}$ equipped ...
-1
votes
0answers
45 views

A question regarding Grothendieck , topos and (adelic??) points

I am having a look at this conference: https://www.youtube.com/watch?v=yNgvvNx_P9w I am particularly interested in getting your feedback on 1:14:30 and the seconds thereafter. Could anyone explain ...
7
votes
4answers
100 views

When $\overline{(a,b)}$ does not equal $[a,b]$

Lee topological manifolds 2.13 c) says For any pair of points a,b in X show that $\overline{(a,b)}\subset[a,b]$. I have done that, but next: Give an example to show that equality need not ...
5
votes
1answer
45 views

Problem in Banach Fixed Point Theorem for a functional equation

I was recently presented this within the context of topological spaces: I am asked to show that there exists a unique continuous function $ f\colon \left[0,\frac{1}{2}\right] \rightarrow \Bbb R $ ...
1
vote
1answer
57 views

what is significant about closed sets? [on hold]

Looking at this, what is the Significance / usefulness of a set being closed? what more can be deduced when a set is proved to be closed ? l am sure it activates number of short cuts in proving ...
3
votes
1answer
25 views

Manipulating the maximum function, metric spaces.

I am trying to show that the supremum metric, $d_{\infty}$, is indeed a metric on $\mathbb R^2$. I have shown that the first two properties of a metric space hold, but am having trouble showing the ...
1
vote
0answers
41 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
votes
1answer
35 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
votes
4answers
66 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
votes
2answers
41 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
vote
1answer
32 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 ...
2
votes
0answers
51 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
votes
4answers
85 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
votes
0answers
14 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
votes
3answers
34 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
votes
0answers
61 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
vote
1answer
24 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 ...
2
votes
1answer
185 views

If a $n$-manifold exists, then is it the boundary of an existing $(n+1)$-manifold?

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
47 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
votes
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
votes
1answer
36 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
votes
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
votes
0answers
12 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
votes
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
96 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
votes
0answers
46 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
vote
0answers
38 views

A mistaken 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 ...
-1
votes
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
votes
1answer
50 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
votes
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
votes
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 ...
4
votes
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
vote
2answers
48 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 ...
4
votes
3answers
148 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 $S^2$. ...
1
vote
0answers
31 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
votes
0answers
52 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
vote
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 ...