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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8
votes
1answer
60 views

Let $A$ be an open set of $\mathbb{R}$ and $B$ any set, under what coniditions of $B$, $AB$ is open?

I don't really know how to establish the conditions so $AB$ can be open. The problem says: Let $A$ be an open set in $\Bbb R$ and $B$ any other set. Define: $$AB = \{xy\in\mathbb{R}\,\colon x\in ...
2
votes
0answers
39 views

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces.

Give an example of a function $f :X \to Y$ which is sequential continuous but not continuous where $X$ and $Y$ are some topological spaces. I have seen some example which uses $X$ to be non ...
4
votes
1answer
90 views

Is $\overline{D}_{\varepsilon}$ a connected Jordan region in $\mathbb{R}^{n}?$

Definition. Let $E$ be a nonempty subset of $\mathbb{R}^{n}$.The distance from a point $\mathbb{x}\in\mathbb{R}^{n}$ to set $E$ is defined by ...
3
votes
0answers
40 views

Computational Topology Codes

I am working on a project with a PI that thinks could be solved with computational topology tools. For this project, we will be looking at the persistent homology of objects in 3D images. I tried ...
1
vote
0answers
9 views

An example of open closed continuous image of $T_2$-space that is not $T_2$

Engelking in his "General Topology" states that $T_2$ separation axiom is not preserved under open closed continuous surjections. In "General Topology" by Stephen Willard I have found two separate ...
2
votes
2answers
39 views

An example of open closed continuous image of $T_0$-space that is not $T_0$

Engelking in his "General Topology" states that $T_0$ separation axiom is not preserved under open closed continuous maps. But I can't find any example of open closed continuous image of $T_0$-space ...
1
vote
2answers
15 views

Cantor's Intersection Theorem

If the subsets of the compact space are already non-empty, isn't it obvious that the even the smallest subset is non-empty, and so the intersection is also non-empty because it would be the smallest ...
4
votes
2answers
34 views

Show that the collection of all open subsets of $X$ that are contained in $Y$ is a topology on $Y$.

This question is from a text book. Please let me know if my proof is vaild. Suppose $X$ is a topological space and $Y$ is an open subset of $X$. Show that the collection of all open subsets of ...
-1
votes
0answers
31 views

Show that $y_n=x_{\phi(n)}$, defines a Cauchy sequence. [on hold]

Let $\phi:\mathbb{N}\to\mathbb{N}$, such that $\displaystyle\lim_{n\to\infty}{\phi(n)}=\infty$. If $(x_n)$, is a Cauchy sequence in the metric space $M$, then $y_n=x_{\phi(n)}$, defines a Cauchy ...
4
votes
1answer
138 views

Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?

So it is weekend! and I am reading a nice book, "The Poincaré conjecture", written by a mathematician (Donal O'Shea, topologist). The book introduces step by step basic concepts of Topology, and talks ...
4
votes
2answers
88 views

Gluing diagrams: is it possible to glue a surface with itself in the same point? how is the diagram drawn?

I am learning the basic concepts of Topology, and playing now with the gluing diagrams (describing the fundamental domain of a topological space), this is an excerpt of a basic description I took from ...
2
votes
1answer
35 views

Looking for a clarification of the Suslin $\mathcal{A}$-Operation with a (finite) example

I have a problem concerning the output of (and the intuition behind) the Suslin $\mathcal{A}$-Operation. More specifically, I really don't see exactly what the output of it really is (even if I can ...
0
votes
0answers
41 views

Homotopic family of curves

I stumbled over the following question. Imagine we have a two homotopic curves on the sphere $\mathbb{S}^1$ namely $\gamma_1,\gamma_2$. Then we can write them as $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
2
votes
2answers
30 views

How to prove that the subsets of $\mathbb{N}$ that don't contain arithmetic progressions of some length form closed sets of a topology?

I have exactly the same problem as this person, which I will rewrite below:Topology and Arithmetic Progressions. The reason I'm posting this is that I'm stuck at a later stage than the OP of that ...
1
vote
1answer
26 views

Is it true that factor spaces are T4 if product space is T4?

I use the following definition of $T_4$-space: for any two disjoint closed sets $A$, $B$ there exist disjoint open sets $U$, $V$ containing $A$ and $B$ respectively. Is it true that factor spaces are ...
4
votes
2answers
171 views

Trying to show that $C([0,1])$ is a complete metric space, using the norm $\|f|| = \max_{x\in [0,1]} |f(x)|$.

I think I have this problem almost done. I am taking $C([0,1])$ to be the set of all continuous function $f\colon[0,1] \to \mathbb{R}$. I have already shown that $\displaystyle\|f\| = \max_{x\in ...
2
votes
1answer
41 views

Help me understand the reasoning used in the following lemma (38.1) from James Munkres' Topology.

Let $X$ be a space and $h: X \to Z$ be an embedding of $X$ in the compact Hausdorff space $Z$. There exists a corresponding compactification $Y$ of $X$ such that $H:Y \to Z$ is an embedding and equals ...
2
votes
1answer
40 views

2.25 of Lee's introduction to topological manifolds

If M is an n-dimensional manifold with boundary, then IntM is an open subset of M , which is itself an n-dimensional manifold without boundary. Here are the definitions to use: If M is an n-manifold ...
-1
votes
1answer
23 views

Continuity in general topological space (non-metric)

When defining continuity using open sets in a general topological space without a metric, is this considered C^0 or C^inf or something in between?
0
votes
0answers
16 views

C^1 mapping of a non-metric topological space - does this make sense?

Is there a way to define a derivative on a mapping between general topological spaces without invoking a metric?
1
vote
3answers
48 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 ...
2
votes
1answer
44 views

subspace of a metric space

Let $(S,d)$ be a metric space, $\mathcal{S}$ the induced topology. $A\subset S$ a subset. It is easy to see that $A\cap\mathcal{S}=\mathcal{A}$, i.e., the topological subspace on $A$ is the ...
1
vote
3answers
41 views

Equivalence of norms problem.

How would I show that $\|\cdot\|_3$ and $\|\cdot\|_\infty$ are equivalent norms on $\mathbb R^2$? I understand that to say two norms are equivalent, then there exist two real constants, $m,M$ such ...
1
vote
0answers
34 views

Examples of generating the same topology

I'm teaching myself topology using a book I found. The question below is from the text. Then there are two additional questions that I am curious about. Please let me know if I'm doing it correct. ...
3
votes
0answers
51 views

Check a proof that, besides $\varnothing$, no open set in $\mathbb{R}^{n}$ has measure zero in $\mathbb{R}^{n}$

I am teaching myself Munkres's Analysis on Manifolds and came across an exercise, stated in the title of this question. Please see my proof below and, if doable, criticize it. That a set has measure ...
6
votes
3answers
196 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 $\mathbb{C}P^{1}\simeq S^{2}$ ...
0
votes
1answer
36 views

How to determinate whether superset will be open or closed?

Let $M = (X, d)$ and A is closed subset of X, i.e. $A \subseteq X$. $A$ is told to be closed, iff it's complement $X\setminus A$ is open in $M$. But how can we determine, whether superset is open or ...
1
vote
2answers
52 views

Why is uncountable union of $\mathbb{R}$ the same as this space

Can anyone give an intuitive reasoning as to why the uncountable disjoint union of copies of $\mathbb{R}$ is the same as $\mathbb{R}$ with discrete topology product with $\mathbb{R}$ with the usual ...
-2
votes
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, ...
0
votes
1answer
24 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 ...
7
votes
3answers
1k views

Banach Tarski — any demonstration?

Is there any where to watch a video of a ball being decomposed into 5 pieces that are then translated and rotated to create two balls? How is this even possible without stretching? Is it possible ...
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 ...
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, ...
-5
votes
1answer
51 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 ...
9
votes
9answers
389 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'} ...
-3
votes
0answers
45 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?
6
votes
1answer
58 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 ...
2
votes
2answers
114 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
1answer
52 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 ...
1
vote
0answers
17 views

Covering a $n$-holed torus for $n\geq 2$ with a hyperbolic tesselation?

How can I cover a $n$-holed torus $(n\ge2)$ with $\frac{2-2n}{\frac pq-\frac p{2}+1}$ faces of regular hyperbolic tesselation {p,q}? I don't need the graphics, just the construction. For example, in ...
0
votes
2answers
73 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
26 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 ...
-2
votes
2answers
46 views

What is meant by “functional analysis is the study of vector spaces endowed with a topology” [closed]

Lecture notes on Functional Analysis by Razvan Gelca open with the definition: Functional analysis is the study of vector spaces endowed with a topology, and of the maps between such spaces. ...
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
3answers
148 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, ...
0
votes
0answers
48 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 $|x_{n}|\leq 1$ ...
5
votes
2answers
223 views

nonstandard topology?

It is possible to have nonstandard models of PA where the natural numbers are different. The definition of a topology requires a notion of finiteness. What happens if we use a nonstandard model ...
2
votes
2answers
50 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 ...
3
votes
0answers
35 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 ...