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
0answers
9 views

$\partial A$, when $A=\{x\in M: f(x)>0\}$ is the set $\{x\in M: f(x) = 0\}$

I have a question about the proof of this fact: $\partial A$, when $A=\{x\in M: f(x)>0\}$ is the set $\{x\in M: f(x) = 0\}$ The proof says the following: $$A = f^{-1}((0,+\infty))$$ Since ...
0
votes
0answers
18 views

Stone-Weierstrass, Urysohn

Let $X$ be a compact space. Show, using Stone-Weierstrass and Urysohn's lemma, that the following statements are equivalent: a) $X$ is homeomorph to a compact subset of $\mathbb{R}^n$ b) It exist ...
1
vote
0answers
6 views

existence of quotient maps to topologically transform one shape into another

What are the mathematical condition that are necessary for a shape A to be transformed into shape B by quotient maps ? For example , I can divide a square into two triangles by drawing a diagonal .Now ...
0
votes
0answers
31 views

What is $S^3/S^1$?

I have been given this space in a question but I am unsure what it means I know that $S^3=\{(z_1, z_2) \in \mathbb{C^2}\mid |z_{1}|^2 + |z_{2}|^2=1 \}$ Could you help me understand what set of ...
1
vote
2answers
17 views

$K$ compact metric space, is there a finite set of continuous functions that separates points in $K$?

Definition: A family of functions $\mathcal{F}$ on a set $X$ separates points in $X$ if for every distinct pair $x,y\in X$ there exists $f\in\mathcal{F}$ such that $f(x)\neq f(y)$. Let $K$ be a ...
-1
votes
0answers
21 views

Can a balloon in flight change its surface topology without tearing? [on hold]

It is very well known that an inflated balloon in flight will not change its surface topology without tearing; the elastic deformations in its surface will be like "smooth transformations" in its ...
0
votes
0answers
32 views

Homeomorphism between $\mathbb{R^2}$ and $S^2-N$, the sphere without its north pole

How would one approach the following problem? Write down a homeomorphism and its inverse from $\mathbb{R^2}$ to the sphere $S^2-N$ without its north pole So I need a function $f(x,y) : ...
1
vote
1answer
21 views

Compact set contained in the interior of another compact set

Let $X$ be a locally compact Hausdorff space. Does the property "every compact set is contained in the interior of some compact set" has a special widely known name? Is it related to paracompactness?
0
votes
0answers
19 views

Computing homology group using Mayer-Vietoris sequence

Suppose I am given an exact sequence: $$0\to G\xrightarrow{f} \mathbb{Z} \xrightarrow{g} \mathbb{Z} \xrightarrow{h} H\to 0 $$ where the first $\mathbb{Z}=H_3(A\cup B)$ and the second ...
0
votes
2answers
38 views

For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N

In order to prove: For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N I'm supposing that $x_n$ is convergent, that is: $$\forall \epsilon>0, ...
0
votes
1answer
34 views

Is this proof regarding product of connected spaces correct?

Let $X,Y$ be connected spaces, and consider their product $X\times Y$. I want to show that their product is connected. The posts I've read here regarding this question often include creating "slices" ...
2
votes
0answers
23 views

Cylinder as trivial fiber bundle with fiber $S^1$?

In prepartion for the example of a Mobius strip, a cylinder is often taken as a first example of a trivial fiber bundle with fiber $I$ and total space $S^1\times I$. However, it seems to me that we ...
0
votes
1answer
37 views

Show thaf $f$ is homeomorphism

Prove that, if $(X,d)$ is a compact metric space and $f:X \to Y$ is continuous invertible then $f$ is homeomorphism. I started like this. Since, $f$ has to be homeomorphism it is equivalent to the ...
0
votes
3answers
47 views

Compactness of $C([0,1])$

I have to verify if the $C([0,1])$, space of all continuous functions defined on interval $[0,1]$ with supremum metric is compact. As I know, we have to check if every sequence of functions ...
0
votes
0answers
22 views

Infinite product probability spaces

Does the infinite product of probability spaces always exist (using the sigma algebra that makes all projections measurable and providing a probability measure on this sigma algebra)? I always ...
1
vote
1answer
23 views

Density in a topological space.

I have the set $X=\{a,b,c,d,e\}$ with topology $T=\{\emptyset, \{a\}, \{a,b\}, \{a,c,d\},\{a,b,c,d\}, \{a,b,e\}, X\}$ I want to identify the dense subsets of $X$. Now, I haven't worked too much with ...
1
vote
1answer
21 views

Why are the (connected) components of a topological space themselves connected?

I am trying to prove that (connected) components of a topological space are connected. I'll first define what I mean by a 'component of a topological space': For a topological space $X$, write ...
0
votes
2answers
26 views

Are slices $\left\{b\right\}\times F\subset B\times F$ homeomorphic to $F$?

Looking at a continuous projection $B\times F\rightarrow B$, are slices $\left\{b\right\}\times F\subset B\times F$ homeomorphic to $F$?
1
vote
1answer
14 views

$A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$

Let A be a real set then is it true that $A$ is measurable if and only if $\forall\epsilon$, $\exists$ open set $G$ and closed set $H$ such that $H\subset A\subset G$ and $\mu(G|H)<\epsilon$.
3
votes
1answer
57 views

Show that $\|f_1+f_2\| \leq \|f_1\| + \|f_2\|$ using Minkowski's inequality

I am trying to show that: $\|f_1+f_2\| \leq \|f_1\| + \|f_2\|$ using the Minkowski inequality. for: $$ \|f\| = \left(\int_0^1 \left[|f|^2 + |f'|^2\right]\ dx\right)^{1/2}.$$ I dont see how I can ...
0
votes
1answer
24 views

Graph of a continuous function $f:M\to N$ is a closed subset of $M\times N$

I need to prove that the graph of a continuous function $f:M\to N$ is a closed subset of $M\times N$. $N$ is a metric space. I think I'm supposed to use this result. So, that's what I did: $Graph(f) ...
0
votes
1answer
18 views

$F=\{x\in M: f_\lambda(x)\ge 0, \forall \lambda\in L\}$ is a closed subset of $M$

I want to discuss this proof that: Let $f_\lambda$ be a family of continuous functions, then: $F=\{x\in M: f_\lambda(x)\ge 0, \forall \lambda\in L\}$ is a closed subset of $M$: Since every ...
1
vote
1answer
21 views

$M=A\cup B$, $f|_A$ and $f|_B$ are continuous, then $f$ is continuous in $A\cap B$

In order to prove: $M=A\cup B$, $f|_A$ and $f|_B$ are continuous, then $f$ is continuous in $A\cap B$ does it suffice to prove: for $a\in A\cap B$: since $f|_A$ is continuous, then $\forall ...
0
votes
0answers
18 views

Continuity in terms of interior of preimage and preimage of interior

Let $f$ be a map between metrix spaces $X,Y$. In order to prove: $f$ is continuous $\iff$ $f^{-1}(\operatorname{Int} Y)\subset \operatorname{Int}(f^{-1}(Y))$ I did: $\rightarrow$ Suppose $x\in ...
-3
votes
0answers
17 views

constructing a manifold structure for a cylinder [on hold]

Any help on this problem would be greatly appreciated. thanks! let M be the cylinder {$(x,y,z)\in \mathbb{R}^3:x^2+y^2=1$} in $\mathbb{R}^3$. Construct a manifold structure each topological space ...
-2
votes
1answer
20 views

constructing a manifold structure for a plane in $\mathbb{R}^3$ [on hold]

Any help on this problem would be greatly appreciated. thanks! Let M be the plane in $\mathbb{R}^3$ with normal vector (a,b,c)$\neq$0. Construct a manifold structure each topological space (M,$\tau$) ...
4
votes
1answer
51 views

Is the unit sphere in product topology a compact set?

Let $A=[0,1]^\mathbb{N}$ endowed with the product topology. I know that $A$ is metrizable and complete, but is the set $\{v\in A: \Vert v \Vert_\infty=1\} $ compact, where $\Vert \Vert_\infty$ is the ...
0
votes
2answers
27 views

Fundamental group of simple graphs

Find the fundamental groups of graphs A, B and C as shown: They look simple but I am unsure what their fundamental groups would be. I was thinking that $A$ and $B$ are generated by just one ...
1
vote
1answer
27 views

How do I know the fundamental group of an infinite graph is well defined?

I get that given a choice of spanning tree and base point for a (connected) graph, I can effectively change the base point through path conjugation, so there's no problem there. For finite graphs, the ...
1
vote
1answer
34 views

Definition of a separable metric space

The book I'm reading doesn't explicitly give a definition of separable metric spaces. The only type of separability definition I know that a separable topological space is one that has a countable ...
0
votes
2answers
26 views

Proving equivalence of statements on continuity between metric spaces

On page 228 of Mícheál Ó Searcóid's Metric Spaces (2007), he writes Criteria for Comparability of Metrics Suppose $X$ is a set and $d$ and $e$ are metrics on $X$. Then the following ...
1
vote
0answers
18 views

Continuously variable *space*

I'm trying to understand formally how and why a fiber bundle with fiber $F$ should be thought of as a gluing of homeomorphic copies of $F$ which varies continuously. I do not understand how this is ...
1
vote
0answers
22 views

Is the Kähler differential of a continuous function ring trivial?

Suppose $A=C^0(\mathbb R)$ is the ring of real-valued continuous functions on $\mathbb R$. Is it true that, the Kähler differential $\Omega_{A/\mathbb R}$ trivial? In other words, suppose that ...
1
vote
1answer
35 views

How does the compactness property help us show a subset $A$ of a metric space $X$ is closed?

We have a compact subset $A$ of a metric space $X$ and we want to show that this implies that $A$ is closed. Let $y \in A$ and $y \in A^c$. For each $y \in A$, we can take open neighbourhoods $U_y$ ...
-3
votes
1answer
35 views

Prove $\bar{A}\setminus \bar{B} \subset \overline{A\setminus B }$

Here is my approach so far Let $A$ and $B$ be subsets of the metric space $(M,d)$ My thoughts on how to prove it, is to choose an element $x$ from $\bar{A}\setminus \bar{B}$ and show it exists in ...
-1
votes
1answer
43 views

Set having a enumerable dense subset [on hold]

I've got an exercise in which I need to prove the following: A set has an enumerable dense subset. Can you share the ways to do this? Edit: X a normed vector space $Y:=B(X,\mathbb{K})$ a vector ...
3
votes
1answer
29 views

Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a ...
0
votes
1answer
24 views

Constructing a metric topology that is the same as the standard topology

Any help on this problem would be greatly appreciated. thanks! $\textbf{Definition:}$ Let $\tau$ be the collection of subsets of $\mathbb{R}^n$ with the following property: $\forall x \in U,\; ...
0
votes
1answer
26 views

Proving that sums of convergent sequences are complete metric spaces

Let $L_1$ be the set of all sequences of real numbers $$x = (x_1,x_2,..., x_n, ...) $$ with the property that $\sum_{n=1}^\infty |x_n|$ is convergent. If we define $$d_1(x,y) = \sum_{n=1}^\infty ...
1
vote
0answers
40 views

Proof: $X$ is Hausdorff if and only if the diagonal $\Delta$ is closed in $X\times X$.

Prove that $X$ is Hausdorff if and only if the diagonal $\Delta$ is closed in $X\times X$. This exercise appeared on a previous exam in my course, and also in Munkres. Here's my attempt: First I ...
2
votes
0answers
22 views

compactness of thes sequence set

Let $S$ be a compact (in the usual topology) subset of $\mathbb R^n$, let $W = \{(q_k)_{k\in\mathbb{N}}\,\mid\, q_k\in S\}$ be the set of all the sequences taking elements in $S$, let ...
0
votes
1answer
30 views

Homotopics curves

We are in the plane (x,y). We have two periodic (closed) planar curves : (x1(t),y1(t)) which is a simple loop and (x2(t),y2(t)) which is a limaçon. Are these two curves are homotopic ?
1
vote
3answers
52 views

Show that the sphere, S, and $\mathbb{R}^2$ is not homeomorphic

I am trying to show that the sphere $S^2$ and $\mathbb{R}^2$ are not homeomorphic.I understand that you can't 'compress' a 3D shape into a 2D plane but I don't know how I would express this formally. ...
1
vote
2answers
43 views

Topology, locally-compact Hausdorff space

I already asked this question here: locally-compact Hausdorff space, equivalent, compact, continuous So if this repost is not apprechiated, please just delete this thread, but I would really like to ...
-3
votes
0answers
46 views

Prove that $[0,1]$ is compact. [on hold]

Prove that $[0,1]$ is compact. Using the definition of compactness. (Not using Heine-Borel theorem)
0
votes
1answer
24 views

Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$

I have the following exercise: Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$ I don't know what $b$ is meant to be, there's a typo in this exercise. I ...
2
votes
1answer
14 views

Proof that a discrete space (with more than 1 element) is not connected

I'm reading this proof that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton ${x}$ and its ...
0
votes
1answer
20 views

$A$ is an open subset of $M$ $\iff$ ($x_n\to a\implies x_n\in A$ for large $n$)

My definition of an open subset $A$ of $M$ is the one that for every $x\in A$, there is an open ball contained in $A$. Now, suppose that $x_n\to a$. By definition, $\forall \epsilon>0$ there exists ...
3
votes
1answer
40 views

Show completeness of metric subspace

I have problems solving the following 2 problems: Given is the metric $d:\Bbb R\times\Bbb R\to[0,\infty[$ with $$d(x,y):=|\arctan(x)-\arctan(y)|\;.$$ a) Show that the metric subspace ...
0
votes
0answers
23 views

Questions about proof of $\lim x_n = a, \lim y_n = b\implies \lim x_n+y_n = a+b$ in a normed vector space

I need to prove that, in a normed vector space $E$, we have: $$\lim x_n = a, \lim y_n = b\implies \lim (x_n+y_n) = a+b$$ and: $$\lim\lambda_n = \lambda, \lim x_n = a \implies \lim \lambda_n\cdot ...