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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5
votes
3answers
681 views

Local homeomorphisms which are not covering map?

I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local ...
0
votes
0answers
25 views

Wot - open set in $B(H)$

To prove an exercise, I need to show if sot - subbase $N(0,h,r) = \{x\in B(H);~~\|xh\|<r\}$, for $h\in H$, is wot - open, then every sot - subbase is wot - open. Definition: $U$ is wot - open if ...
3
votes
1answer
128 views

Planar kelvin problem

What is the minimal possible value of the maximal total side length shared by any two tiles in a tiling of the plane if all tiles have the same area $A$? $\text{Total side length} = ...
3
votes
1answer
19 views

Lower Limit Topology Properties

I am reading topology from Munkres book. While reading the countability and Separation axioms, I came across several references to Lower limit topology ($\mathbb{R}_l$) which essentially comprises of ...
8
votes
1answer
661 views

Braid groups and the fundamental group of the configuration space of $n$ points

I am giving a lecture on Braid Groups this month at a seminar and I am confused about how to understand the fundamental group of the configuration space of $n$ points, so I will define some ...
0
votes
1answer
46 views

how shall i'll prove if c={(x_n) :exists lim x_n}is a Hyperplane, dense, or closed? [on hold]

Let $$c=\{(x_n) :\exists ~ \lim x_n\},$$ where $c$ is included in $\ell^\infty$. How can I find a function $T$ such that $\ker(T)=c$? Also, after that, how can I see if $T$ is continuous? Thanks.
0
votes
0answers
19 views

Complex Projective Space as a Quotient of a Disc

I am reading Hatcher's book and I have a problem understading how the complex projective space $\mathbb CP^n$ can be realised as a quotient of $D^{2n}$ (page 7) Let me briefly outline his arguments ...
-1
votes
2answers
45 views

Is this sequence is dense?

Define $S _m, _n = $ n th smallest square number which is bigger or same than $10^ {m-1}$and smaller than $10^m$ Then is the sequence $ \frac{S_m,_n} {10^m}$ is dense in (0,1) or arbitary ...
1
vote
1answer
25 views

Prove (in the example) that being homotopic depends on the range of the Homotopy

Question: Define $F : [0,1]\times [0,1] \rightarrow X$ by $F(x, t) = (cos(\pi x), (1 - 2t) sin(\pi x))$. Take a straight-line homotopy between $F(x, 0)$ and $F(x, 1)$. Show that they are ...
2
votes
1answer
89 views

Prove that $x+g$ is homeomorphism

Problem: Assume we have $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ of $C^1$ class with derivative bounded uniformly by some constant $M<1$. Consider ...
0
votes
1answer
51 views

Open ball metric space vs open set topological space

I'm having trouble understanding the notion of an open set when applied to a space without a metric defined on it - I have read that all metric spaces are naturally a topological space, but the ...
0
votes
0answers
32 views

Is the unit square a $2$-manifold in $\mathbb{R}^2$?

I'm using the following definition of a (smooth) manifold: It's from J.Munkres "Analysis on Manifolds". This is an exercise taken from this book: Is the unit square $[0,1]\times [0,1]$ a ...
0
votes
1answer
25 views

$f$ proper but not universally closed

Say that a continuous function $f$ is universally closed if $f \times 1_T$ is closed for all topological spaces $T$, and call a function proper if inverse images of compact sets are compact. I know ...
2
votes
1answer
39 views

Why is a simply connected 3-manifold a homotopy 3-sphere?

I recently looked at the statement of the Poincare conjecture, and realized I didn't know why the fact that a 3-manifold is simply connected implies that it is homotopic to a 3-sphere. Could someone ...
2
votes
0answers
28 views

Prob. 2 (a), Sec. 26 in Munkres' TOPOLOGY, 2nd ed: Is every set compact in the finite-complement topology?

Here's Prob. 2 (a), Sec. 26 in Topology by James R. Munkres, 2nd edition: Show that in the finite-complement topology on $\mathbb{R}$, every subspace is compact. I think I can show this. Now ...
6
votes
1answer
53 views

Characterization of open and closed maps in terms of filters

Understanding continuity and compactness in terms of filters has been very clarifying for me. Is there a convenient characterization of open and closed maps in terms of filters? For instance, it ...
4
votes
1answer
49 views

Munkres, Chapter 2, question on locally finite family of sets

I've been working through the Munkres Topology text on my own, and I am not sure if the following argument is correct. Fishing around the internet a bit for some alternative answers and it looks like ...
1
vote
1answer
22 views

How to prove that a Banach space of analytic functions containing $H^\infty$ except the origin is simply connected?

If $X$ is a Banach space of analytic functions on the unit disk $D$ which contains the space of analytic bounded functions on $D$, how can I prove that $X\setminus\{0\}$ is simply connected?
17
votes
2answers
285 views

Completion of the real numbers

On the real line $\mathbb{R}$ endowed with euclidean topology i may put different metrics, inducing the same topology, but inducing different completions. For example if one considers the standard ...
0
votes
0answers
24 views

Cantor Set in Alexander Horned Sphere Construction

I have seen it said in several different places that in the standard construction of the Alexander horned sphere, given by successive embeddings of a sphere with $2^n$ handles, either limited or ...
0
votes
1answer
29 views

Closure of an interval in specific topology

Given a topology constructed on elements of the type $$ (p,q] $$ where $p$ and $q$ are rational numbers. Let us call this topology $\tau_{-,Q}$, given an interval of type $(a,b]$ its closure in the ...
1
vote
1answer
80 views

why separable normal space has only continuum many different open subsets?

I cannot prove the fact in the title. Please help! I am reading the handbook of set theoretic topology. And I found this fact in a proof in the book. Thank you.
0
votes
1answer
20 views

Diffeomorphism between covering spaces

Let $\pi_1: M \rightarrow M_1$ and $\pi_2: N \rightarrow M_2$ be two smooth covering maps. Now $\phi: M \rightarrow N$ is a smooth diffeomorphism. Does this induce a smooth diffeomorphism $f: M_1 ...
4
votes
5answers
61 views

Explanation of $\overline{\lim} A_n$ and $\underline{\lim}A_n$

Let $(A_n)_n$ be a countable family of subsets of a set $X$. We define: $$\lim \inf A_n = \underline{\lim} A_n = \bigcup_{n \in \mathbb N} \bigcap_{k \ge n} A_k$$ $$\lim \sup A_n = \overline{\lim} ...
1
vote
1answer
31 views

Homeomorphism of the closed unit ball not preserving the sphere?

Exercise 2.9.12 in Ronnie Brown's Categories and Groupoids asks the reader to show that if $f:\mathbb{R}^n \to \mathbb{R}^n$ is continuous such that $f$ restricts to a homeomorphism from the open ...
0
votes
1answer
28 views

Outer Regularity of the Lebesgue measure on the Hilbert brick

Is the product measure on the Hilbert brick $I=[0,1]^\mathbb{N}$ outer regular (that is measure of every set is the inf of measures of open sets, containing it)?
4
votes
0answers
55 views

Properties of $\mathbb{C}P^n$

I'm currently working on a somewhat deformed version of $\mathbb{C}P^2$ and want to check some properties from a geometrical and/or topological point of view. Of course, $\mathbb{C}P^2$ is Kähler ...
1
vote
1answer
14 views

Open condition given by inequality on functions

Let's say we have two functions $f,g\in C^\infty(D)$, $D$ an open domain in $\mathbb{R}^2$. The condition $f(x,y)<g(x,y)$ is an open condition on $D$? With this I mean: do the points $(x,y)\in D$ ...
13
votes
5answers
2k views

Why are box topology and product topology different on infinite products of topological spaces?

Why are box topology and product topology different on infinite products of topological spaces ? I'm reading Munkres's topology. He mentioned that fact but I can't see why it's true that they are ...
2
votes
2answers
33 views

Kernel of a bounded linear operator on a normed linear space need not be closed or open?

How should be the kernel of a bounded linear operator on a normed linear space as a set? Kernel of a bounded linear operator on a normed linear space need to be closed or open? Or it need not be ...
0
votes
0answers
31 views

Connectedness of circle without center line across it

Using a definition I saw in an old Russian book, a set in $\mathcal R^{n}$ is said to be connected if it cannot be represented as a disjoint union of two nonempty, separated sets. Separated, meaning ...
3
votes
1answer
34 views

How do I prove the converse of Stone-Weierstrass theorem?

Let $X$ be a locally compact Hausdorff space. Let $\bar \rho$ be the uniform metric on $\mathbb{R}^X$ and $\mathscr{A}$ be an $\mathbb{R}$-subalgebra of $C_0(X,\mathbb{R})$ which is dense in ...
1
vote
3answers
43 views

How to determine the closure of a subset and prove it is actually the closure?

I have this subset $E = \{r \in Q: r^2 \leq 2\}$ which is in $\mathbb{R}$ with the Euclidian metric. I was wondering how can I find the closure of this subset. Here is what I have: The limit points ...
1
vote
2answers
42 views

Finding limit points for these sets

Here's my resoning for finding limit points for some sets. Could you guys read it and see if it's all good? <3 $$\{(x,y)\mid \ x^2+y^2<1\}$$ For this set, its kinda simple to see that every ...
1
vote
1answer
74 views

Finite mapping $f : \mathbb R^2 \to \mathbb R$

Is there an continous function $f: \mathbb R^2 \to \mathbb R$ such that $f^{-1}(a)$ is finite for every $a \in \mathbb R$? It's not possible for analytic or smooth but I'm curious about continous ...
0
votes
1answer
20 views

Prove that a complete field defines a partition of a set

Let $\Omega$ be arbitrary set. Let $Q$ be a partition of $\Omega$. I already proved that the collection of all unions of the cells in $Q$ is a complete field $\mathcal{F}$ (complete field is ...
0
votes
1answer
45 views

What does arbitrary mean in the following context?

I recently encountered a theorem stating, "The arbitrary product of compact spaces is compact". What does arbitrary product mean in this context? Any product of compact spaces? If that is the case why ...
1
vote
0answers
78 views

Distance between sets

Let $K \subset K_1 \subset U \subset \Bbb R^2$, such that $K$ and $K_1$ are compact sets, with $K \subset \mathring {K_1}$, and $U = \mathring U \subsetneq \Bbb R ^2$. If $w \in \partial K_1$ such ...
3
votes
1answer
43 views

Surjectivity of $\mathcal{id}_{\mathbb{R}^n}+g$ when $g$ is a contraction?

Assume $g:\mathbb{R}^n\longrightarrow \mathbb{R}^n$ is a contraction and consider $h=\mathcal{id}_{\mathbb{R}^n}+g$. The map $h$ is injective. Is it always surjective? My question has the following ...
1
vote
0answers
23 views

What is the formalism for a map that returns the adjacent vertex positions of a given adjacency matrix?

How do I formally denote a map that returns the adjacent vertex positions of a given adjacency matrix? Example: V = undirected adjacency matrix $$ V = \left[ \begin{matrix} v_{1,1} & v_{1,2} ...
1
vote
0answers
14 views

Proposed proof for quasi-metric result

A quasi-metric on a set $X$ is mapping $\rho: X \times X \rightarrow [0, \infty)$ satisfying the following conditions: $\rho(x,y) \geq 0~~\text{and}~~\rho(x,x) = 0;$ $\rho(x,z) \leq \rho(x,y) + ...
3
votes
1answer
63 views

Reconciling two different definitions of constructible sets

This question is really about sets and topology, but it is motivated from commutative algebra, hence the tag. Setup: Let $X$ be a set and let $\{U_\lambda\}_{\lambda\in\Lambda}\subset 2^X$ be a ...
3
votes
1answer
40 views

Is this space Hausdorff and are these two spaces homeomorphic?

Let $S^1 = \{(x,y) \in \mathbb{R^2} \ | \ x^2 + y^2 = 1 \}$. We define an equivalence relation $\sim$ on $S^1$ such that $(x,y) \sim (x',y')$ if and only if $y = y'$. Now we study the following ...
0
votes
5answers
58 views

Finite set of points of $R^n$ is compact

In order to show that a finite set of points of $R^n$ is compact, I just need to show that the set is closed and bounded. First of all, since it's a finite set, I can Always pick the greatest ...
0
votes
0answers
24 views

Cartesian product between manifolds

I was given the following exercise: Show that if $M$ is a $k$-manifold without boundary in $\mathbb{R}^m$, and if $N$ is an $l$-manifold in $\mathbb{R}^n$, then $M \times N$ is a $k+l$ manifold in ...
2
votes
1answer
24 views

Kelly's topology: Compact spaces and finite intersection property

Lemma A topological space is compact iff each family of closed sets which have the finite intersection propertyhas a non-void intersection. I've proved the same result in another way but i really ...
0
votes
0answers
33 views

When is a connected space, path connected?

Let $X$ be a connected topological space. When is $X$ path connected? Is the Hausdorff property enough? Is it too much?
0
votes
2answers
19 views

limit points of subset of real numbers

Let $$A=\{ \frac{\sqrt{m} -\sqrt{n}}{\sqrt{m}+\sqrt{n}} | m,n\in \Bbb{N} \}$$ I think that we must find sequence of $A$ and find limit of sequence,let $a_m =\frac{\sqrt{ k ^2 m^2} -\sqrt{ ...
0
votes
0answers
62 views

Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$ [on hold]

Let $f\in C^{\infty}(Ω)$ for some open set $Ω \subset R^n$ that contains $0$. Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$. I found this problem in a ...
1
vote
1answer
73 views

Spectral Measures: Polar Decomposition

Isometric Equality Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^{**}$$ It gives rise to operators: ...