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 (3)

1
vote
0answers
35 views

Grassmannian Manifold homeomorphism

I have troubles understanding the meaning of this excercise: I am supposed to show that the Grassmannian manifold $G_{k,n}$ of k-dimensional subspace in $\mathbb{R}^n$ is homeomorphic to $O(n)/(O(k) ...
1
vote
0answers
25 views

symmetric quasi-uniformity

A quasi-uniformity $U$ will be called symmetric provided that $U = U^{-1}$, that is, provided that it is a uniformity. Otherwise it will be called nonsymmetric. It is readily seen that the supremum ...
1
vote
0answers
38 views

Continuity of a function in the product topoogy

Hi everyone I would like to understand if my reasoning is correct. Let $X$ be the space of sequences with values in the interval $[0,1]$, i.e. if $\mathbb{N}$ is the set of natural numbers, $x\in X$ ...
1
vote
0answers
33 views

Can a factor map be a Serre fibration?

Let $D_n$ be an $n$-disc. Is the factor map $p: D_n\to D_n/S^{n-1}\simeq S^n$ a Serre fibration, in other words, can any homotopy $F: [0,1]\times X\to S^n$ be lifted to $\tilde{F}: [0,1]\times X\to ...
1
vote
0answers
22 views

$\omega$-covers and $S_1(\Omega,\Gamma)$ property

I am readund this article and there is some proof in there (top of page 156) which is not clear to me. The definitions are: 1. Property ($\gamma$): If $\mathcal U$ is an $\omega$-cover of $X$, ...
1
vote
0answers
69 views

Sets which are open “modulo a nullset”

A set $A$ is said to have property of Baire there exists an open set $U$ such that $A\triangle U$ is meager. So this says that symmetric difference of $A$ and some open set is small (in the sense of ...
1
vote
0answers
44 views

Let $X$ a finite set, and $X^{*}=X\cup \{\omega\}$ wiht $\omega\notin X$. Given a filter $\mathcal{F}$ on $X$

Let $X$ a finite set, and $X^{*}=X\cup \{\omega\}$ wiht $\omega\notin X$. Given a filter $\mathcal{F}$ on $X$, Show that $$\mathcal{T}(\mathcal{F}):=2^{X}\cup\{F\cup \omega\mid F\in\mathcal{F}\}$$ ...
1
vote
0answers
43 views

If a basis is also a subbasis for the same topology, is it closed under finite intersections?

I'm trying to understand a little bit the relation between a basis and a subbasis for a given topology. So suppose $S$ is a subbasis for a topology, say, $\delta_s$. Suppose that $S$ is also closed ...
1
vote
0answers
67 views

Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
1
vote
0answers
108 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
1
vote
0answers
26 views

The game $G(K,X)$

In Telgarsky - Topological games, in page 246, the following game $G(K,X)$ is described: There are given a space $X$ and a class $K$ of spaces such that $Y \in K \Rightarrow \mathcal F(Y) \subset K$. ...
1
vote
0answers
25 views

Completeness of moves for polygonal knots

I am going through the paper, MINIMAL KNOTTING NUMBERS, by MANN et. al. On page six of the paper, they defined following moves for polygonal knots. Parallel moves Triangular moves I understand ...
1
vote
0answers
54 views

Locally compact space .

How can I show that for every locally compact space $X$ there exists a one-to-one continuous mapping of $X$ onto a compact space. And is it necessary that $X$ is compact since we know that every ...
1
vote
0answers
44 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
1
vote
0answers
31 views

Neighborhoods for continuous functions between CG spaces

I have a couple of problems regarding the existence of certain neighborhoods, so as to prove continuity of suitable functions. Suppose then that $Y,X$ and $Z$ are compactly-generated Hausdorff spaces ...
1
vote
0answers
30 views

Does any $\omega$-cover in which $X \in L(\mathcal U)$ is also a $\gamma$-cover?

As a continuation to this question: An $\omega$-cover, is an open cover $\mathcal U$ of $X$, such that, $X \notin \mathcal U$, and for every finite set $F \subset X$, there exists an open set $U ...
1
vote
0answers
51 views

Topology in Infinite Galois Theory.

I am a final year undergraduate student in Mathematics. I have a good background in algebra up to Galois theory of finite extensions of fields. I have started trying to understand the Galois theory of ...
1
vote
0answers
39 views

$\overline{A}=A\cup \delta A$ proof

Can somebody help me out with proving the following equality? $\overline{A}=A\cup \delta A$ where $\delta A=\overline{A} \cap \overline{A^c}$.
1
vote
0answers
36 views

Topology - Compactness of $\mathbb{Z}\times\{0,1\}$

A question from my h.w.: Is the topological space $\mathbb{Z}\times\{0,1\}$ (where $\mathbb{Z}$ has the discrete topology and $\{0,1\}$ the trivial one) compact? sequentially compact? ...
1
vote
0answers
82 views

Hanging a picture with Beta functions

There's a classic puzzle that goes something like this: You have two nails in a wall, and you want to hang a picture with a string (think of a necklace with a pendant) in such a way that if you ...
1
vote
0answers
46 views

What is wrong with this proof in topology?

Let $X$ be a $T_4$-space and $M \subset X$, then $M$ is a subspace that is also $T_4$. Proof: If $A,B$ are closed in $M$, then $A=W_A \cap M$ and $B = W_B \cap M$ for some closed sets $W_A,W_B$. ...
1
vote
0answers
36 views

Compactification via embeddings and extending continuous functions

My question comes from reading Munkres' Topology, the section on Stone-Čech compactification. To find the compactification $\mathrm{Y}$ of $\mathrm{X}$, we find an embedding h, $\mathrm{h}: X ...
1
vote
0answers
28 views

Can we say that $[0,\omega_1]$ is strictly-Frechet with no winning strategy in $G_{np}(q,E)$?

Let $E$ be a topological space, $q \in E$. The neighbourhood point game $G_{np}(q,E)$, is defined as follows. It is played by two players, ONE and TWO.In the n's step $n \in \omega$, ONE chooses ...
1
vote
0answers
40 views

Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
1
vote
0answers
77 views

Nagata Smirnov Metrization Theorem

I am looking for a proof for Nagata-Smirnov Metrization Theorem, but I couldn't find one that is readable. I found the paper by Nagata written in 1954 but it is unreadable and uses old notation. ...
1
vote
0answers
50 views

topological equivalence on interior of $D^2$ that is not continously extendable to $D^2$

As said in the title, I'm trying to find a topological equivalence on the interior of $D^2$ that is not continously extendable to $D^2$. I have an idea about this, so here it goes: Let ...
1
vote
0answers
29 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 ...
1
vote
0answers
20 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 ...
1
vote
0answers
28 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 ...
1
vote
0answers
63 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
0answers
16 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 ...
1
vote
0answers
41 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 ...
1
vote
0answers
32 views

How to show that these two constructions of Tychonoff product topology are equivalent?

Definition: The Tychonoff product topology on $X = \Pi_{t \in T}X_t$, is the topology $\tau$, which is generated by the family $\bigcup\{ p_t^{-1}(\tau(X_t)) : t \in T \}$ as a subbase where ...
1
vote
0answers
96 views

n-Torus with antipodal points identified

If we have n-torus $S^1 \times S^1 \times S^1 \times ....$ n times, and $\mathbb{Z}_2$ acts on this just sending each component of $S^1 \times S^1 \times S^1 \times ....$ to its antipodal. What will ...
1
vote
0answers
28 views

understanding topological argument in rado-kneser theorem

Rado-kneser choquet theorem states that Poisson integral of a homeomorphism of unit circle is a homeomorphism. It's proof goes like proving it local homeomorphism by proving non vanishing of jacobian ...
1
vote
0answers
59 views

An example of a topological space which is a group, but is not a “topological group”

Is there any example of a topological space $X$ which has a group structure, but the maps $(x,y)\mapsto xy$ and $x\mapsto x^{-1}$ are not continuous?
1
vote
0answers
21 views

About a decreasing sequence of convex domains

$\newcommand{\int}{\operatorname{int}}$ Let $\Omega_n$ a decreasing sequence of open, bounded and convex domains in $R^n.$ Define $$\Omega = \int \left(\overline{\bigcap \Omega_k}\right)$$ I am ...
1
vote
0answers
50 views

Amount to choice necessary to prove instances of Tychonoff theorem

Let $I$ be a fixed nonempty set. I would like to know how much choice is necessary in order to prove that the product of any $I$-indexed family of compact topological spaces is compact (under the ...
1
vote
0answers
38 views

Open Book Decompositions of 3-manifold and Associated Heegard Splittings

In page 13 of the paper: http://arxiv.org/pdf/math/0510639v1.pdf It is stated that "An open book decomposition (S,h, K) , gives rise to a special Heegard decomposition of M ". Here, S is a surface , ...
1
vote
0answers
155 views

Wedding Vows puzzle

My father came up with a puzzle and dared me to solve it. I could solve it by trial and error, but I rather want to solve it mathematically. It is the so called "Wedding Vows puzzle" where you have to ...
1
vote
0answers
52 views

Fundamental groups of configuration spaces

In a previous answer see here by Samuel Reid, I read the following: "The configuration space of $n$ points in a topological space $X$ is usually defined to be, $$C_{\hat{n}}(X) = \{(z_{1},...,z_{n}) ...
1
vote
0answers
40 views

What “topological setting” would make complex analysis fluent?

In measure theory, the order topology on $\overline{\mathbb{R}}$ (extended real) and $[0,\infty]$ provides rich foundation to analyze measurable functions and abstract integral. Just like this, i ...
1
vote
0answers
26 views

Does an homeorphism function maps an interval in X to an interval in Y?

Let $(X,\leq_X)$ and $(Y,\leq_Y)$ be two well-ordered sets. Let $f:X\rightarrow Y$ be an homeorphism between $X$ and $Y$ and their order topology (relative to $\leq_X$ and $\leq_Y$). My question is, ...
1
vote
0answers
47 views

Why is convergence in measure topologizable?

I'm aware that pointwise convergence and uniform convergence are topologizable since the former can be made by seminorms and the latter with a norm. I'm also aware that pointwise a.e. fails because ...
1
vote
0answers
51 views

caracterization of lower semicontinuous functions

Let $X$ a topological space satisfying the first contability axiom. I want to prove the following result: $\varphi : X \rightarrow R $ is lower semicontinuous (this mean that $\varphi^{-1} (a, + ...
1
vote
0answers
73 views

Limit Point vs Boundary Point

I'm reading Kosniowski's book on algebraic topology, and I have a question about how he defines limit points. He says that for a subset $Y$ of a topological space $X$, the limit points of $Y$ are ...
1
vote
0answers
39 views

A question about rectifiable curves

Does every rectifiable curve that is a subset of the Euclidean plane have zero two-dimensional Lebesgue measure?
1
vote
0answers
44 views

Showing a piecewise function is continuous.

Given the following path defined on some topological space $H(s,t) := \begin{cases} f(s,t) , \space K(s) \leq L(t)\\ g(s,t) ,\space K(s) \geq L(t)\\ \end{cases} $ I wish to prove that $H$ is ...
1
vote
0answers
36 views

dimension of a subspace of a flag variety

Let $X$ be a topological space. If $X = \bigcup U_\alpha$ is an open covering of $X$ then $$\dim X = \sup_\alpha \dim U_\alpha.$$ Now suppose that $X = \coprod U_\alpha$, i.e., $X$ is the disjoint ...
1
vote
0answers
30 views

Generating sets of topology on $C_b(\mathbb T)$ with supremum Norm?

Can someone tell me what sets generate the topology on the continuous bounded function on the 1-dim Torus $C_b(\mathbb T)$ with the supremum norm? Are the sets $\left\{ f \in C_b(\mathbb T) : ...