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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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 ...
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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 ...
2
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0answers
37 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 ...
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0answers
30 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 ...
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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 ...
6
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1answer
56 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 ...
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39 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 ...
2
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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 ...
2
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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 ...
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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 $\gamma_{i}(t) = e^{i \alpha_i (t)}$ ...
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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 ...
2
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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
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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 ...
2
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1answer
39 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 ...
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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?
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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?
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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 ...
4
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1answer
89 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 ...
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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. ...
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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 ...
2
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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 ...
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1answer
35 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 ...
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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 ...
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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, ...
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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 ...
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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, ...
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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 ...
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1answer
50 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 ...
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0answers
44 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?
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1answer
57 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 ...
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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 ...
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2answers
72 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 $S$.
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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 ...
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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'} ...
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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.
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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 ...
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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 ...
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0answers
47 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
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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, ...
3
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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 ...
2
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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 ...
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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 ...
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0answers
52 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 ...
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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
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0answers
44 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 ...
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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} ...
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1answer
14 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 ...
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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 ...
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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. ...
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3answers
41 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 ...