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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11
votes
1answer
277 views

Extending open maps to Stone-Čech compactifications

Let $X$ be a Čech-complete space, and $Y$ a paracompact space. Suppose $f\colon X\to Y$ is a continuous and open surjection. Since $Y$ is completely regular we have that $\beta(Y)$ is homeomorphic to ...
6
votes
2answers
563 views

How to prove that this set is closed?

Suppose $Y$ be an ordered set in the order topology. Let $f, g: X \to Y$ be continuous. How to show that the set $\{x| f(x) \leq g(x)\}$ is closed? This is a excercise from munkres. Maybe trying to ...
3
votes
3answers
205 views

Why are these two expressions equivalent?

Let $X$ be a topological space and let $A\subseteq X$. Can someone please tell me 1) what the difference between these two expressions is 2) if they are equivalent (if they aren't can someone give me ...
1
vote
2answers
129 views

How should I measure the total “closeness” of a finite number of elements?

Suppose I have n points and a way to measure the pairwise (probably non-Euclidean) distance between them. I would like to have some way to measure the total "closeness" of my points, but I'm not ...
3
votes
2answers
177 views

Chain of closed subsets in separable metric space

I need some help to prove that if $\mathcal{A}$ is a chain of closed subsets in a separable metric space then there is a countable subfamily $\mathcal{A}'\subseteq\mathcal{A}$ such that ...
5
votes
0answers
192 views

Does this property of scattered spaces have a name?

Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{(\alpha)}$ the $\alpha$th derivative of $K$ by the Cantor-Bendixson derivation (i.e., define ...
1
vote
1answer
252 views

applications of topology or abstract algebra to astronomy

That is my question, there are applications of the branch of topology or abstract algebra to the astronomy? I know that there are to physics but to astronomy?
4
votes
2answers
392 views

Closed curves on the discrete torus

I came about the following graph which seems to me the smallest discrete version of the torus: Is this graph treated under a special name? What can be said about its cycles? Can its cycles be ...
0
votes
1answer
125 views

Relation between a map and its lifting into the covering space

I have the following question: Let $\mathbb{D}$ denote the unit disk. Let $f:X_1 \longrightarrow X_2$ be a continuous mapping between Riemann Surfaces. Let $ \pi_1 : \mathbb{D} \longrightarrow X_1$ , ...
3
votes
1answer
162 views

Fiber bundle M x M - diagonal

Under what conditions for a space $M$ does the projection map to the first factor $p: M \times M - \Delta \rightarrow M$ has the local triviality condition, i.e. is a fiber bundle? Where $\Delta$ ...
25
votes
1answer
796 views

What is the cardinality of the set of all topologies on $\mathbb{R}$?

This was asked on Quora. I thought about it a little bit but didn't make much progress beyond some obvious upper and lower bounds. The answer probably depends on AC and perhaps also GCH or other ...
2
votes
2answers
110 views

Topologies on spaces of mappings

Given two topological spaces $X, Y$, the only example I know of a topology on the space $\mathcal C(X,Y)$ of continuous mappings from $X$ to $Y$ is the compact-open topology. However I presume that ...
3
votes
2answers
504 views

Uses of Lebesgue's covering lemma

Consider Lebesgue's covering lemma in the following form: Let $(X,d)$ be a compact metric space and let $\{U_i\}_{i\in I}$ be an open cover of $X$. Then there exists $\delta>0 $ such that each ...
0
votes
2answers
81 views

If $Y$ has irreducible components $Y_1, \cdots, Y_n$, then the $\overline{Y_i}$ are the irreducible components of $\overline{Y}$

Let $X$ be a noetherian topological space, $Y$ a subspace having irreducible components $Y_1, \cdots, Y_n$. Prove that the $\bar{Y_i}$ are the irreducible components of $\bar{Y}$. I think as the ...
6
votes
1answer
2k views

Prove $\epsilon$-$\delta$ definition of continuity implies the open set definition for real function

I need to prove that the $\epsilon$-$\delta$ definition of continuity implies the open set definition continuity for a real function. Here's my attempt. For any basis $V: (a, b)$ in the range, for ...
0
votes
1answer
100 views

Is this union finite?

Suppose for a compact topological space(if you want, we can assume Hausdorff) $X$, $X$ is a disjoint union of compact subsets,that is $X=\bigcup_{i\in I} X_i$ such that $X_i\cap X_j=\emptyset$ for ...
4
votes
1answer
274 views

Verifying that these sets form a topology

I am solving Exercise 4.1, Question 17(v) from Topology without Tears (link) by Sidney Morris. (This exercise is marked with a star.) Let $S = \{ \frac{1}{n} \,:\, n \in \mathbb N \}$. Define a ...
3
votes
0answers
293 views

Useful topology on space of smooth structures on $\mathbb R^4$?

Mathoverflow is intimidating, so I thought I'd ask here first (second). If I don't get any useful answers here in a few days, I'll ask there. $Q_0$: Is there any use for a topology on the (continuum ...
5
votes
1answer
230 views

Paracompact image of an open continuous map from a Čech-complete space

Theorem: Assume that $X$ is Čech-complete, and $Y$ is paracompact. If there exists $f\colon X\to Y$ which is surjective, open and continuous then $Y$ is Čech-complete. The theorem appears ...
2
votes
3answers
498 views

Why is $[0,1]^{[0,1]}$ not first countable?

In a ps file on topological properties, the set $[0,1]^{[0,1]}$ is given as an example of a product topology that is not first countable. Is there a proof of why?
2
votes
2answers
124 views

Countable subfamily with union equal to that of the containing family

Recently it was explained here that in a second countable topological space $X$, any base admits a countable subfamily which is also a base. I know a base $\mathcal{B}$ covers the space $X$, so ...
2
votes
2answers
236 views

Bases having countable subfamilies which are bases in second countable space

I don't understand a proof right at the beginning of this document found here. This proof is on why any base for the open sets in a second countable space has a countable subfamily that is a base. ...
7
votes
2answers
504 views

“Proof” that $\mathbb{R}^J$ is not normal when $J$ is uncountable

In 'Topology' by Munkres, he leaves as an exercise to prove that $\mathbb{R}^J$ is not normal under the product topology when $J$ is uncountable. The proof outlined as exercise 32.9 is the same one ...
2
votes
3answers
207 views

If $f^{-1}(B)$ is compact for all compact $B$, then $f(A)$ is closed for all closed $A$

How to prove this exercise? Let $f$ be a continuous function of $\mathbb R^n$ to $\mathbb R^m$, so that $f^{-1}(B)$ is compact in $\mathbb R^n$ for all compact $B$ in $\mathbb R^m$. Prove that ...
2
votes
1answer
131 views

Embedding a manifold in the disk

I don't understand a sentence made by Hirsch in his Differential Topology at page 175: If $k > n+1$ and $M^n = \partial W^{n+1}$, then an embedding $M^n \hookrightarrow S^{n+k}$ extends to a neat ...
1
vote
0answers
67 views

connected sum of $n$ copies of $\mathbb{RP}^2$

$n\mathbb{RP}^2$ is $2n$-gon with identified edges. Can you check directions of arrows in picture? Thanks.
2
votes
1answer
119 views

Closed path on torus

How to proof that closed path $l$ (see picture) on 2-dimensional torus don't homotopic to trivial path, using definition torus as CW-complex? (closed path on surface $S$ is a map $f:[0, 1]\to S$, such ...
2
votes
1answer
289 views

Continuous vector field on surface

On plane we can easy to define continuous vector field: $$\mathbb{R}^2\ni x\longmapsto v(x)\in\mathbb{R}^2,$$ where real-valued function $v_1(x)$ and $v_2(x)$ are continuous. But when we try to ...
1
vote
1answer
248 views

$\{ (x,y) \in \mathbb R^{2} | x>0, y \in \mathbb R \}$ not clopen?

Let $S_{1} = \{ (x,y) \in \mathbb R^{2} | y \geq \frac{1}{x}, x> 0 \}$ and $S_{2} = \{ (x,y) \in \mathbb R^{2} | x = 0, y \leq 0 \}$. Now $S_{1} + S_{2} = \{ (x,y) \in \mathbb R^{2} | x > 0, y ...
3
votes
2answers
141 views

The fundamental group of $K_{3,3}$ — relationship between its generators and embedding into manifolds

So I've been reading this wonderful PDF textbook on algebraic topology: http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf In particular, I'm very interested in the chapter on graphs. This ...
2
votes
1answer
208 views

Example to illustrate the necessity for nets?

Given topological spaces $(X,\tau)$, $(Y,\upsilon)$, we say a function $f:X\rightarrow Y$ is continuous iff $\forall$ $U\in \upsilon$, $f^{-1}(U)\in \tau$. Equivalently, we can say that $f$ is ...
1
vote
1answer
110 views

Curves on the projective plane

I have two little questions, I'm learning this, and I'm not accustomed yet )=. The questions are so simple. First define on $$ R^3 - \left\{ {\left( {0,0,0} \right)} \right\} $$ the topology given ...
0
votes
2answers
290 views

Homeomorphic maps in the Euclidean space with the Euclidean metric

I am currently reading an introduction to topological and metric spaces and want to know whether the following statement is true: Consider the Euclidean space $\mathbb{R}^n$ endowed with the ...
4
votes
3answers
830 views

(Explicitly) Constructing Deformation Retractions

I'm having trouble building the actual deformation retractions, although I understand the concepts behind them, homotopies, etc. For example, when constructing a deformation retraction for ...
5
votes
1answer
487 views

Motivation for the term “separable” in topology

A topological space is called separable if contains a countable dense subset. This is a standard terminology, but I find it hard to associate the term to its definition. What is the motivation for ...
0
votes
1answer
235 views

How can I prove that these two topologies are the same?

In this case, I'm only confused, sorry for ask, but I'm learning the quotient topology ._. I know that the problem it's trivial . Sorry again for ask. Let $ S^2 $ and define an equivalence relation ...
6
votes
1answer
633 views

vector field on sphere with two handles

Consider a sphere with two handles. If I don't make a mistake it is torus with one handle: Can you give me an example of vector field on it with one singular point. Thanks.
1
vote
1answer
174 views

Confusion about compactness in Counterexamples in Topology

On page 19 of Counterexamples in Topology, the authors say "sequential compactness clearly implies countable compactness" without explaining. I feel dumb, why is this so obvious? Does anyone have ...
5
votes
2answers
283 views

Question on proof that first countable, countable compact space is sequentially compact

I'm reading some notes with a proof that a first countable, countably compact space is sequentially compact. On page 118 of these notes (third page of the pdf), they are constructing a convergent ...
3
votes
1answer
147 views

how can I prove formally that this set is not path connected

Let $ \mathbb{R}^2 $ with its usual topology, let $D$ the set of all the lines that pass through the origin, with rational slope. And add to $D$ some point that does not lie in any of the lines ( call ...
4
votes
0answers
230 views

Weakest topology with respect to which ALL linear functionals are continuous

One often considers a Banach space $X$ under the "weak topology", ie. the weakest topology such that all bounded linear functionals are continuous. This leads me to wonder about the weakest topology ...
1
vote
1answer
121 views

Basis for topology of weak convergence of probability measures

Let $S$ be a metric space with be Borel $\sigma$-algebra $\Sigma$. Let $\boldsymbol{P}(S)$ be the set of probability measures on $(S,\Sigma)$. According to wikipedia, "the weak topology is generated ...
5
votes
1answer
496 views

Quotient Topology

Lets $X$ is a topological space and $Y$ is some subset of $X$. How we define topology of quotient space $$X/Y=\{x\in X~|~x\sim y\Leftrightarrow x, y\in Y\}.$$ Thanks.
4
votes
2answers
345 views

Space-filling curve with distance locality

Is there a space-filling curve $C$ that has the property that, if $C$ passes through $p_1=(x_1,y_1)$ at a distance $d_1$ along the curve, and through $p_2$ at $d_2$, then if $|p_1 - p_2| \le a$, then ...
1
vote
1answer
299 views

Principal and fiber bundles as defined by Husemoller

In his book 'Fiber Bundles' Husemoller defines principal bundles and fiber bundles quite differently from how they are usually defined. Specifically: Definition: a right $G$-space $X$ is called ...
1
vote
1answer
111 views

Different topologies $\Rightarrow$ different neighborhood bases?

Can someone give me a proof or a counterexample to the following: Given two different topologies over the same set, the neighborhood bases of these two topologies have to differ in at least one point. ...
7
votes
1answer
556 views

Completely metrizable implies $G_\delta$

It is a consequence of Lavrentyev's theorem that a metrizable space is completely metrizable if and only if it is a $G_\delta$ subset in every completely metrizable space containing it. In my ...
6
votes
2answers
2k views

Is Topology an important class to take before Functional Analysis?

I am starting a graduate degree in math pretty soon and I am planning to take a course in Functional Analysis and Spectral Theory. Topology is being offered next semester as well but I don't think it ...
4
votes
3answers
308 views

Is it always possible to simply expand a simple 2D polygon with any point?

Given a simple 2D polygon P = ( M1 .. Mn ) and a point M, is it always possible to construct a new simple polygon P' by "adding" M to P as a new vertex? If so, is this always possible without ...
8
votes
2answers
177 views

In/out equivalent to left/right “chirality”

Apologies if this is off-topic, but we're having a problem over on English Language with this question, and I thought you guys might be able to help. Basically it's a matter of topology. We know the ...