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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-2
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1answer
51 views

Axiomatic proof that all points of an open set are interior points

In "Principles of Mathematical Analysis, Rudin the following definition (f) to open sets: a set is open if all of its points are interior points Sidney Morris' Topology Without Tears, however, ...
2
votes
3answers
50 views

Proving that a set is open using epsilons.

I'm trying to prove that the set $$A=\{x=(x_{1},x_{2})\in\mathbb{R}^2:x_{1}^{2}+x_{2}^{2}>1\}$$ is open in $\mathbb{R}^2$ with the usual norm is open with the definition of "epsilons". My attempt ...
0
votes
1answer
14 views

Why is a convex subspace the requirement for equivalence beween subspace and order topologies?

I'm currently studying topology, and in one of the lectures we were presented with a theorem that went something like this (rephrasing since I don't have the theorem in front of me): Let $(X, ...
0
votes
1answer
31 views

Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other

Let's consider following spaces: $L^2(\mathbb{R}) = L^2(\mathbb{R}, \mathbb{C}, \mu_L)$ --- space of $\mathbb{C}$-valued functions defined on $\mathbb{R}$ for which the square of the absolute value ...
-5
votes
1answer
52 views

cauchy sequence in metric space [on hold]

Can you tell me an example of a function from an metric space $(X,d_1)$ to an metric space $(Y,d_2)$ s.t image of every cauchy sequence in $X$ is a Cauchy sequence in $Y$ but $f$ is not uniform ...
-3
votes
0answers
49 views

about cauchy sequence in metric space [on hold]

Let $f$ be a function from a metric space $(X,d_1)$ to a metric space $(Y,d_2)$. If the image of every Cauchy sequence in $X$ is a Cauchy sequence in $Y$, how can I prove that $f$ is continuous?
8
votes
1answer
62 views

Two disjoint real projective planes in real projective space?

Let $\mathbb{R}\mathbb{P}^3$ be the real projective three-space. It is clear that any two hyperplanes in $\mathbb{R}\mathbb{P}^3$ intersect. But I wonder whether one could embed two copies of the real ...
1
vote
1answer
27 views

What can we say about open unit balls of sup-norm and integral-norm

Consider the normed linear spaces $X_1=(C[0,1], ||.||_1)$ and $X_{\infty}=(C[0,1],||.||_{\infty})$ , where $C[0,1]$ denotes the vector space of all continuous real valued functions on $[0,1]$ and ...
0
votes
2answers
76 views

Example of a set $S$ that is countable, but the set of limit points is uncountable [on hold]

What would be an example of a set $S$ so that $S$ is countable. However $S'$ is uncountable. In this $S'$ is the set of all the limit points of $S$.
0
votes
0answers
28 views

Circle topologically different from a line interval, torus from a rectangle (proof)

I am new to topology, so I request a proof of this intuitively simple concept, so I can start getting a grasp of the subject. I have heard the argument: If we remove any point from the circle, it ...
9
votes
9answers
391 views

Motivation for the Definition of Compact Space

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} ...
1
vote
1answer
45 views

What do you call a space whose only compact sets are finite? [duplicate]

What do you call a topological space where a subset is compact iff it's finite? Is there a technical name? For example, take the discrete topology, or the countable complement topology.
0
votes
1answer
66 views

Manifold that is not a Euclidean space

I just started reading a textbook, and it keeps saying that an $n$-dimensional manifold is a topological space with the same local properties as Euclidean $n$-space. I don't really understand what is ...
2
votes
2answers
115 views

What is the “topology induced by a metric”?

My book gives the following definition: Let $(M,d)$ be a metric space, and let $\mathcal{T}$ be the collection of all subsets of $M$ that are open in the metric space sense... $\mathcal{T}$ is ...
0
votes
0answers
49 views

Hilbert Cube and Metric Space

Given that $d(x,y)=\sum_{n=1}^{\infty}2^{-n}|x_{n}-y_{n}|$ defines a metric on $H^{\infty}$ where $H^{\infty}$ is the Hilbert Cube, a collection of all real sequence $x=(x_{n})$ with $|x_{n}|\leq 1$ ...
2
votes
3answers
157 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
37 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
51 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
57 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
67 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
49 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
15 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
45 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
14 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
27 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
29 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
429 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
27 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
54 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
32 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
34 views

Prove or disprove about isomorphic functions [closed]

Prove that : if f is an isomorphic then it is continuous or not?
1
vote
1answer
50 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
38 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
53 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
33 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
175 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
206 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
157 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 ...
4
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
27 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 ...