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; ...

learn more… | top users | synonyms (2)

1
vote
2answers
37 views

Topologically dense subgroup

Let G denote the group of orientation-preserving isometries of the plane; equivalently, the group of affine transformations of the complex field C of the form $z \rightarrow \alpha z + \beta$ ...
1
vote
0answers
18 views

Question on HSP and SHPS inquality.

In the screenshots attached above George Bergman outlines his way of proving $HSP \ne SHPS$ I understand the first definition as the group of affine transformations and each element of the group ...
24
votes
3answers
359 views

Are there surfaces with more than two sides?

I'm watching a naive introduction to the Möbius band, the lecturer asks if it's possible to construct a one sided surface and then she says that there is one of these surfaces, namely the Möbius band. ...
4
votes
1answer
43 views

Constructing paths which are not constant on any interval

Suppose $X$ is a Hausdorff topological space (or metric space if you like) and $f:[0,1]\to X$ is any non-constant path. It could be that $f$ is constant on a closed interval $[a,b]$, and it is ...
0
votes
0answers
37 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
1
vote
1answer
28 views

Property of normal spaces

Let $X$ be a normal space and let $U_1$ and $U_2$ be open subsets of $X$ such that $X= U_1 \cup U_2$. Show that there are open sets $V_1$ and $V_2$ such that $\operatorname{cl}(V_1) \subseteq ...
1
vote
0answers
38 views

How can we characterize all topological groups given $G$?

The idea is that all topologies on G (not necessarily making it a topo group) can be completely specified by a set of functions $F = \{f: G \to G\}$ if you form a basis for the topology like: $B = ...
1
vote
3answers
63 views

Is convex hull of a finite set of points in $\mathbb R^2$ closed?

Is the convex hull of a finite set of points in $\mathbb R^2$ closed? Intuitively, yes. But not sure how to show that. Thanks!
0
votes
0answers
32 views

Why is the vertex called non-manifold vertex?

I am working on triangle meshes in one 3D reconstruction project for a while. I know what one manifold vertex looks like and how to detect them. But I hope to understand the definition of non-manifold ...
1
vote
0answers
15 views

Neccessary and sufficient conditions to form a topological ring on $\Bbb{Z}$?

Let $B = \{ \{a + b f_i(n) : n\in \Bbb{Z}\} : a,(b\neq 0) \in \Bbb{Z}, f_i \in F \}$. Then what are necessary and sufficient conditions on the set of integer functions $F$ such that $B$ is a basis ...
2
votes
2answers
49 views

Derivatives on Functors

I'm not even sure if this question makes pedantic sense but is there any way to rigorously define the notion of taking the derivative of a functor?
1
vote
1answer
61 views

How to show that $\mathbb{Q}_p^*$ is totally disconnected?

Let $\mathbb{Q}_p$ be the field of p-adic numbers and $\mathbb{Q}_p^*$ the set of invertible elements in $\mathbb{Q}_p$. How to show that $\mathbb{Q}_p^*$ is totally disconnected? Thank you very ...
1
vote
1answer
50 views

If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$?

Let $X$ be a Hausdorff topological space, $A$ subset of $X$. Let $I$ be an interval $[0,1] \subset {\mathbb R}$. Suppose that $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ ...
1
vote
0answers
26 views

All topology pairs $(X,Y)$ such that $f: X \to Y$ is continuous.

Given an arbitrary function, or more specifically if you want let $R$ be a ring and let $X = S \times S; Y = R; S \subset R$ and $f(a,b) = a - b$, is there something interesting about all the topology ...
4
votes
1answer
36 views

Lindelöf if and only if every collection with the countable intersection property has non-empty intersection of closures

I am trying to study for my topology exam, and my professor recommended this question from the text (Munkres's Topology (2nd edition), Section 37 question 2): A collection $\mathcal{A}$ of subsets of ...
0
votes
1answer
35 views

Question on Quotient spaces [closed]

(i) Show that the quotient space $(S^{2} \times [0,1])/(S^{2} \times \{0\})$ is homeomorphic to the 3-disc $D^{3}=\{(x,y,z)\in \mathbb R^{3} \mid x^{2}+y^{2}+z^{2}\leq 1\}$ (ii) Let ...
1
vote
0answers
35 views

Product Topology & Homeomorphic [closed]

(i) Describe the topological space $S^{0}$ x $S^{0}$ Can I just say $S^{0}$x$S^{0}$={(-1,1)x(-1,1)}? I know the definition of a product topology is saying that XxY has a basis consisting of all ...
5
votes
1answer
47 views

Compact topological space not having Countable Basis?

Does there exist a compact topological space not having countable basis? I have constructed a product space from uncountably many unit intervals $[0,1]$, endowed with the product topology. ...
0
votes
1answer
28 views

A simple question on Hausdorff distance

Let $(A_n)$ be a sequence of compact sets in $R^n$ and consider $K$ and $A$ compact sets in $R^n$. Suppose that $A_n \cup K \rightarrow A \cup K$ in the Hausdorff distance. Then $$ A_n ...
1
vote
0answers
34 views

Why is proof of the [topological] closed graph theorem incorrect?

Specifically, the closed graph theorem I am referring to is: Let $f : X \rightarrow Y$ exist and $Y$ be compact and Hausdorff. Then $f$ is continuous if and only if the graph of $f$ denoted by $G_f = ...
1
vote
1answer
47 views

Demonstrating that the Mandelbrot Set is connected

I know that demonstrating the Mandelbrot Set is connected requires a non-trivial proof, and that Mandelbrot himself was fooled at first. But can it be demonstrated visually that the set is connected? ...
3
votes
1answer
101 views

A proof about $F_\sigma$, $\sigma$-compact sets, and subsets of the irrationals

I've been looking at a proof that shows the following result. $\mathbb{P}$ is the set of irrational numbers, $\mathbb{Q}$ the rationals, and $\mathbb{R}$ the reals. The following conditions are ...
0
votes
1answer
24 views

Mapping Class Group of $S^3$

I am wondering if we can compute $\pi_0(Homeo(S^3))$ (i.e. the group of hoemomorphisms of the three-sphere mod isotopy) or if anyone has a reference where I could find such information.
5
votes
1answer
113 views

What exactly is a dimension?

Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a ...
1
vote
1answer
29 views

Is the following proof of: $X = [0,\omega_1]$ does not satisfy $S_1(\Omega,\Gamma)$ correct?

Definitions: An open cover $\mathcal U$ of $X$ is called a $\gamma$-cover, if for every $x \in X$, the set $\{ U \in \mathcal U : x \notin U \}$ is finite. An open cover $\mathcal U$ of $X$ is ...
0
votes
1answer
37 views

being connected of R^infinity?

is $\mathbb R^\infty$ ($\mathbb R\times \mathbb R\times \mathbb R\times \mathbb R\times\cdots$) connected with metric $P$? $P(x,y)=\sup\{D(x(k),y(k))\mid k\in\mathbb N\}$ for all $i$: ...
1
vote
2answers
84 views

How is $\mathbb R^2\setminus \mathbb Q^2$ path connected?

Prove $(\mathbb R$ x $\mathbb R)-(\mathbb Q$ x$ \mathbb Q)$ is path connected. I know I need to let $(x_0, y_0), (x_1, y_1) \in$$(\mathbb R$ x $\mathbb R)-(\mathbb Q$ x$ \mathbb Q)$ and then consider ...
2
votes
1answer
23 views

Does parR imply Souslin?

I have encountered the following property in this article: We say that a space $X$ parR (partition-Rothberger), if, for every sequence $(\mathcal P_n : n \in \omega)$, of partition of $X$ into ...
1
vote
1answer
22 views

How to prove that any product of separable spaces has the Souslin property

A topological space $X$ has the Souslin property if every pairwise disjoint family of non-empty open subsets of $X$ is countable. I am trying to solve the following exercise: Prove that any product ...
2
votes
3answers
59 views

Is $\mathbb{Q}^2$ connected?

Is $(\mathbb Q \times \mathbb Q)$ connected? I am assuming it isn't because $\mathbb Q$ is disconnected. There is no interval that doesn't contain infinitely many rationals and irrationals. But ...
2
votes
1answer
23 views

Is set on lower-limit topology path-connected?

Is $\mathbb R$ endowed with the left-hand topology (also called lower limit topology) path-connected? Intuitively, I know that the answer is yes but I'm not sure how to prove it. Would it suffice to ...
2
votes
1answer
29 views

Is R with finite complement top path-connected?

I need to prove whether $\mathbb R$ with the finite complement topology is path-connected or not. Is the following proof valid? The function $ g:(\mathbb R,u) \rightarrow (\mathbb R,fc) $ is ...
1
vote
0answers
13 views

Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
3
votes
1answer
63 views

Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
0
votes
3answers
59 views

Homotopy on the unit circle

I am trying understand why the identity function on the unit circle $X=\{(x,y): x^2+y^2=1\}$ is not homotopic to $f: X \to X$ where $f(z)=(1,0)$ for all $z\in X$.
0
votes
1answer
78 views

Prove that a closed ball is closed

Prove that in $\[\vec{E}\]$ normed vector space $\[B(\vec{x}, \varepsilon )\]$ is a closed set. and $\[B'(\vec{x}, r)\]$ is an open set. Fo the first part I created a sequence ($\[x_{n}\] $) ...
0
votes
4answers
46 views

Simply connectedness in $R^3$ with a spherical hole?

I understand why $R^3 - {(0,0,0)}$ is simply connected, and I also understand why $R^2 - {(0,0)}$ is not simply connected. The way I look it at is if checking if the region is $a)$ path-connected and ...
1
vote
0answers
34 views

Comparison of two final topologies

Consider the vector space $F$ of all infinite sequences of reals numbers, such that only finitely many terms of each sequence are nonzero. I recently encountered an exercise where I was required to ...
0
votes
1answer
35 views

Non First Countable: Lack of Information by Sequences [closed]

Can someone proof that when considering sequences in non first countable spaces information will be lost... I'm thinking of sth like there is necessarily a subset whos closure properly contains the ...
0
votes
1answer
27 views

problems with proving that f and g are homotopic.

i need to give an example of 2 continuous functions $f,g: X \rightarrow Y$ which are not homotopic, with: $X = [0,1] \times [0,1]$ and $Y = [0,1] \cup [2,3]$ and i need to show how many homotopical ...
0
votes
0answers
12 views

Calabi homomorphism of the disk

There is a fact that the homomorphism $Diff_0^{\infty}(\mathbb{D},\partial\mathbb{D},area)\to \mathbb{R}$ is surjective, we can use Calabi homomorphism to prove it, where ...
1
vote
2answers
30 views

strong topology = inductive limit topology on duals of projective limits

I've been bothering with this for some time now, and can't find any source with an actual proof, the statement simply appears to be "well-known". If you know (a source with) a proof, I'd be happy :) ...
1
vote
1answer
63 views

Need help understanding this proof about Gelfand spectrum

Consider the following theorem: Let $A$ be a complex non-unital commutative Banach algebra and let $\Omega (A)$ denote its Gelfand spectrum / character space. Then $\Omega (A)$ is locally compact. ...
0
votes
1answer
61 views

Is there a name for the one-point compactification of $\mathbb{C}$?

Let $\hat{\mathbb{C}}$ be the one-point compactification of $\mathbb{C}$. This space $\hat{\mathbb{C}}$ is called the Riemann sphere. If I want to designate the topology $\tau$ on ...
0
votes
1answer
36 views

What does 'real-valued' function mean in topology?

If I have a topological space $X$ and a 'real-valued' function $f$ on $X$. Does this mean I have a map of the form: $f: X \rightarrow \mathbb R$ where $\mathbb R$ has the usual topology? Or something ...
4
votes
2answers
398 views

How to think about a homeomorphism?

Two disjoint circles in the Euclidean space are homeomorphic to two circles interlocked without touching each other. My professor said that to a topologist they are the same thing. I don't understand ...
2
votes
1answer
37 views

Using Cantor's intersection theorem

Assume $f: X \rightarrow X$ is a continuous map where X is a compact metric space. Prove that there exists a non-empty set $A \subset X$ such that $f(A) = A$. (Hint: Set $F_1 = f(X), F_{n+1} = ...
0
votes
0answers
22 views

let $X$ be compact and $p:X\to X/R$. Prove that $X/R$ is hausdroff iff $p$ is closed.

Please refer me which book I should follow for answer. Let $X$ be compact and $p:X\to X/R$ prove that $X/R$ is hausdroff iff $p$ is closed.
2
votes
0answers
29 views

How to show that the Tychonoff product is associative?

Let $\{ X_t : t \in T\}$, be a family of topological spaces. Suppose thst $T = \bigcup \{ T_s : s \in S \}$, where $T_s \neq \emptyset $ for all $s \in S$, and $T_s \cap T_{s'} = \emptyset$ if $s ...
2
votes
1answer
42 views

Lie Automorphisms

Take $X$ to be a Lie group. Define a Lie automorphism of $X$ to be a group isomorphism from $X$ to itself which is also a homeomorphism. Define $Aut(X)$ to be the group of Lie automorphisms of $X$ ...