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)

2
votes
3answers
174 views

Are strongly equivalent metrics mutually complete?

Maybe I'm missing something, but I can't seem to find any references to my exact question. If two metrics, $d_1(x,y)$ and $d_2(x,y)$ are strongly equivalent, then there exists two positive constants, ...
3
votes
0answers
39 views

Coconvergent topology basis?

Consider the space $X=\{\frac1n:n\in\mathbb N_+\}$, with the "coconvergent topology": $$\mathcal O=\{A:(A=\varnothing)\lor(\sum_{x\notin A}x<\infty)\}$$ That is, a nonempty set is open iff its ...
2
votes
2answers
52 views

Show that the following mapping is a contraction.

I have the following problem from a past paper: "Show that the mapping, $$T(x_1,x_2)=\left(\frac{x_1+2x_2}5-1,\frac{x_1-2x_2}7+1\right)$$ is a contraction on $(\mathbb R^2,d_\infty)$." I ...
2
votes
1answer
44 views

How to prove that space is not connected

I found a definition that the space $M$ is not connected if there are open subsets $A,B$ such that $M=A\cup M,A\ne\emptyset\ne B,$ and $A\cap B=\emptyset$. How can I prove from the definition that ...
2
votes
0answers
58 views

What theorems or frameworks explain why the roots of well-behaved functions $h : \mathbb{R} \leftarrow \mathbb{R}^2$ seem to be made up of “pieces”?

First, some terminology: given functions $g,f:Y \leftarrow X$, the equalizer of $g$ and $f$ is defined to be the set of all solutions $x \in X$ to the equation $g(x)=f(x)$ in $Y$. Okay. The following ...
1
vote
2answers
68 views

How to determine whether those sets are open or closed?

Given those three sets below, A (left), B (center) and C (right), with A, B, C $\subseteq \mathbb{R^2}$, how can I determine, whether they are open or closed in metric space terminology via simplest ...
2
votes
0answers
54 views

QFT and topology

I have had a course in topology, I have heard of homotopy quantum field theory and topological field theory, but I dont know anything about QFT, what would be a good starting point to learn about the ...
0
votes
2answers
40 views

Show a linear $\mathbb{R} \rightarrow \mathbb{R}^2$ function is continous

I'm trying to get a sold foundation on my understanding of topological continuity, so I want to make sure I can accurately prove some simple examples. Show that the function $f: \mathbb{R} ...
1
vote
1answer
18 views

How to check F:AxI->B is continuous

A and B are topological spaces.Let f and f' are continuous maps from A to B and homotopic.Then we need F:AxI->B,continuous,where F(s,0)=f(s) and F(s,1)=f'(s). Now my question is if we want to ...
1
vote
1answer
17 views

Intersection of a dense set with an open set is dense in the open set

Let $A\subset M$ ann open subset, of the metric space M. If $X\subset M$ is dense in M, then $X\cap A$ is dense in A. My approach: If $X\subset M$, and $A\subset M$ is a open subset. Let be ...
1
vote
2answers
34 views

Are there any interesting non-metrics whose open balls generate a topology?

Let $X$ be some set. I am wondering if there are any interesting functions $\rho: X \times X \to \mathbb{R}$ whose open balls are the base for a topology on $X$, and where $\rho$ is not a metric (e.g. ...
2
votes
3answers
47 views

When the set of $r$-far interior points from a set is open

Let $E$ be a subset of a metric space $X$ and for $r > 0$ let $$ E_r = \lbrace x \in E : d(x,E^c) > r \rbrace .$$ Is the set $E_r$ always open? Equivalently, is the function $ x \mapsto ...
2
votes
2answers
72 views

Show that $f(x) = mx+b$ is continuous

I'm trying to get a sold foundation on my understanding of topological continuity, so I want to make sure I can accurately prove some simple examples. Show that the function $f: \mathbb{R} ...
1
vote
1answer
16 views

subsets in projective system

Let $\{Z_i\}$ a directed projective system of quasi-compact topological spaces with projective limit $Z$. Assume we are given open subsets $U_i \subseteq Z_i$ such that: 1) For every $i \leq j$, the ...
1
vote
1answer
33 views

Closed subspace of a metric space in which distance between any two points is at most $1$

$X$ be a metric space and $Y$ be a closed subspace of $X$ such that distance between any two points is at most $1$. Then $1$. $Y$ is compact $2$. Any continuous function from ...
0
votes
1answer
30 views

Prove complete metric space for $I=]0,\infty[$ with $d(x,y)=\lvert\ln(x)-\ln(y)\rvert$ [duplicate]

Let $I=]0,\infty[$ equipped with the metric $d(x,y)=\lvert\ln(x)-\ln(y)\rvert$, $\forall x,y \in I$. Prove that $(I,d)$ is complete. Any help, and thanks in advance.
1
vote
2answers
26 views

Problem in showing that a norm is a norm on one space, but not on another.

I have the following question from a past paper: "Consider the two sets, $$A=\{g\in C^1([0,1]):g(0)=g(1)=0\}$$ and, $$B=\{g\in C^1([0,1]):g'(0)=g'(1)=0\}$$ both subsets of the vector space ...
3
votes
2answers
432 views

Is it possible to define Cauchy sequences in a topological space?

I know that we can define Cauchy sequences in topological vector spaces. How about in general topological spaces? Is it possible to define a Cauchy sequence in general topological spaces?
0
votes
0answers
35 views

Ball shape on different metric spaces and interior set

I am starting to study topology, and to assess if I am on the right track, I kindly ask if someone can check my reasoning below. Let the metric space $(\mathbb{R}^2,d)$, where $d$ is the Euclidean ...
2
votes
0answers
66 views

Existence of a Rectifiable Piecewise Smooth Path

Suppose you have $\gamma(t):[0,1]\rightarrow \mathbb{C}$ simple piecewise smooth, $\gamma(0) = 0$ and $\gamma(1)=1$. Does there exist $\eta:[0,1]\rightarrow \mathbb{C}$, another simple piecewise ...
-1
votes
1answer
33 views

Topology induced by discrete metrics and topology induced by singleton [closed]

Show that the topology generated by singleton sets is topology induced by discrete metric. $$d(x,y)= \begin{cases} 0,&\text{if } x=y\\ 1,&\text{if } x \ne y\\ \end{cases} $$
-4
votes
1answer
35 views

Prove or disprove about isomorphic functions [closed]

Prove that : if f is an isomorphic then it is continuous or not?
1
vote
1answer
53 views

The cone of a topological space is contractible (why is the homotopy well defined?)

If $X$ is a topological space, define ${\rm Cil}(X) = X \times I$ the cylinder over $X$, and the cone over $X$, $\operatorname{Con}(X) = \operatorname{Cil}(X)/{\sim}$ the quotient by saying that ...
4
votes
1answer
40 views

Ways to link the unknot to a pole

Is there a way to show that the following ways of linking an unknot to an infinite horizontal pole are inequivalent? Perhaps the Wirtinger presentation would work, but I am not sure because of the ...
0
votes
1answer
55 views

topology (upper limit and lower limit)

I have to show that upper limit topology and lower limit topology on $\mathbb{R}$ (Real line) are not comparable. But suppose if we take $[a,b)$ and $(a-1,b]$, where $a-1 > a$, then isn't it ...
2
votes
2answers
34 views

Mobius over the sphere is the sphere itself

The Mobius band can be thought as a line bundle over $S^1$ by giving the vector spaces half a twist at some point. Now, we can do the same kind of construction by considering a line bundle over the ...
11
votes
1answer
183 views

Is paralellizability a topological invariant (Invariant under homemorphism)

This MO post is a motivation to ask: Is paralellizability a topological invariant (Invariant under homeomorphism)?
6
votes
3answers
217 views

How can one compare these two 4-manifolds

We would like to compare the following two real 4 dimensional manifolds: 1)$M$=The tangent bundle of $S^{2}$ 2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$ ...
2
votes
3answers
163 views

Average of points on an xy plane

I was at a family reunion yesterday which required a bit of travel. Most of that part of the family lives near one another, so I am the outlier. I can't reasonably expect them to have the next reunion ...
3
votes
2answers
76 views

Showing that a rectangle is equal to the closure of its interior

I'm trying to show that if Q is a rectangle, then Q equals the closure of Int Q. I have that the closure of Int Q is a subset of Q and I'm now working to show that Q is a subset of the closure of Int ...
0
votes
1answer
23 views

How would I make continuous functions to form these sets? Parametarizing of sets

How would I make continuous functions to form these sets?(So the domain is connected) I need continuous functions that map connected sets to these in question. $1. \text{Cone}$ $$(x,y,z)| \ ...
3
votes
0answers
28 views

Proving a version of the Kronecker's Theorem

I am trying to prove the following version of the Kronecker's Theorem: Suppose $k$ is a positive integer and $\{1, \theta_0, \dots, \theta_{k-1}\}$ is linearly independent over $\mathbb Q$. Then ...
1
vote
2answers
28 views

continuous map of connected set is connected, example: Proving the connectedness of this set.

I thought I would try to use this to prove connectedness in this set if possible: $$\{(x,y)\mid 1<x^2+y^2<4\}$$ $f(x,y)=x^2+y^2$ So since $(1,4)$ is connected in $\mathbb R$ so it this set, as ...
3
votes
0answers
21 views

What are the modes of vibration of a genus-2 surface?

So it's spherical harmonics for a sphere. The vibrations of a torus presumably are just ordinary string harmonics around each loop. But what are the harmonics on a genus-2 surface (a donut with 2 ...
-2
votes
1answer
46 views

How can I prove that $(X,τ)$ is a Hausdorff topological space?

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1\times X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff ...
0
votes
0answers
53 views

Trying to prove that a continuous function $\,f: \mathbb{D}^2 \rightarrow \mathbb{R}^2$ has a zero using the fundamental group?

Prove: For a continuous $\;f: \mathbb{D}^2 \rightarrow \mathbb{R}^2,\;$ $\,f\left(\omega,0\right) \notin f\left(S^1\right),\,$ given a non constant function $\left(f_{\mid S^1}\right)_{\ast} : ...
2
votes
2answers
68 views

For $l^2 (\mathbb{N})$ is $(l^2, d_2)= (l^2, d)$ topologically, where $d$ forms the usual product topology on $\mathbb{R}^{\mathbb{N}}$?

If we define $d(x,y)= \max_{n \in \mathbb{N}}( \min \{2^{-n}, |x_n - y_n| \})$ then does this distance form the same topology on $l^2 ( \mathbb{N})$ (the set of all square-summable real sequences) as ...
1
vote
1answer
49 views

Discontinuous $ f : \mathbb R^2 \to \mathbb R$ with unusual topology on $ \mathbb R$

With the usual topology on the reals $\mathbb R$ , let $D$ be the family of dense open sets and let $T=D \cup \{ \phi \}$. Let $S$ be the set $R$ with the topology $T$ on it. Show that the function ...
6
votes
1answer
208 views

What type of surface is it?

The picture shows Sphere, Torus, Klein Bottle and Projective Plane, respectively: What about the following one? Is it also Projective Plane? : PS inside triangles and color in shapes are ...
0
votes
1answer
31 views

If $X, Y$ are topological spaces, does $A \times B$ closed in $X \times Y$ imply $A$ closed in $X$ and $B$ closed in Y?

Suppose $X, Y$ are topological spaces, and $A\subset X$, $B\subset Y$. Does $A \times B$ being closed in the product space $X \times Y$ (with the product topology) imply that $A$ is closed in $X$ and ...
1
vote
0answers
23 views

A question uniform convergence

Let $X$ be a compact Hausdorff space, $a$ a continuous real-valued function on $X$, and for $t\in\mathbb{R}$ let $f_t(x)=\exp(ia(x))$ such that the function $t\mapsto f_t$ is continuous (where we use ...
1
vote
1answer
35 views

If a compact subset is contained in an open subset in $\mathbb{R}^n$, is a small cylinder of this compact subset also contained in the open set?

Let $O\subseteq \mathbb{R}^n$ be an open set, $K\subseteq \mathbb{R}^{n-1}$ a compact set and $a\in \mathbb{R}$, such that $$\{a\}\times K\subseteq O$$ holds. Does there exist an $\epsilon>0$, ...
1
vote
1answer
38 views

Example of a separable, locally-compact metric space which is not $\sigma$-compact

I am looking for an example of a separable, locally-compact metric space which is not $\sigma$-compact. At first I thought I could show that if a metric space is separable and locally-compact, then ...
1
vote
2answers
57 views

How does convergence imply continuity?

I'm trying to develop some background understanding to eventually prove the following: ....................... Let $M$ and $N$ be metric spaces and let $f : M \rightarrow N$ be a map. Show that $f$ is ...
0
votes
1answer
42 views

Prove $S^1\times I\to D^2,(s,t)\mapsto ts$ is an identification.

First of all, every map here is continuous. I'm trying to prove the following: If every $S^1\to X$ is homotopic to a constant map $S^1\to X$ then every map $S^1\to X$ extends to a map $D^2\to X$ ...
4
votes
2answers
136 views

Prove that closed subsets of a compact set is compact. What's wrong with this proof?

I understand other methods of achieving the result, but this was my first try. I'm not sure where my mistake is, if any. And yes, I realize that using the fact that $B$ is closed would help. For a ...
5
votes
1answer
99 views

Do there exist general conditions underwhich we can conclude that continuity on a topological space is detected by $\mathbb{R}$?

Whenever $X$ is a topological space, let us say that continuity on $X$ is detected by $\mathbb{R}$ iff for all functions $f : X \rightarrow Y$ where $Y$ is another topological space, we have that if ...
1
vote
2answers
65 views

Prove: If $f: X \subset \mathbb{R}^n \rightarrow Y$ has a continuous extension to all $\mathbb{R}^n$ then $f_\ast$ is trivial.

Prove: If $f: X \subset \mathbb{R}^n \rightarrow Y$ is continuous and has a continuous extension to all $\mathbb{R}^n$ then $f_\ast$ is trivial. I'm not sure how the fact that there exists an ...
3
votes
1answer
88 views

Can a fractal be a manifold?

Here it is said that it is not possible: Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower? But I am confused about this. What about the invariant ...
1
vote
0answers
33 views

Sufficient condition for a infinite countable or non-countable intersection of open sets is equal to an open set.

Let $(X,\tau)$ a no discrete topological space. If necessary for an affirmative answer consider a metric space $(X, d )$ or a Banach space $(X, \|\,\cdot\, \|)$. In these cases, the topology $\tau $ ...