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)

2
votes
0answers
83 views

Is there a $P$-space linearly Lindelöf and non-Lindelöf?

A completely regular topological space $(X,\tau)$ is a $P$-space, if every $G_\delta$-subset of $X$ is open (i.e $\tau$ is closed under countable intersection). A topological space $X$ is linearly ...
2
votes
0answers
45 views

Is this proof correct: domain of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected.

The domain $X$ of a quotient map $p:X\rightarrow Y$ is connected when $p^{-1}(y)$ is connected for each $y\in Y$ and $Y$ is connected. Proof: If $X = F \uplus G$ for two nonempty closed sets $F,G$ ...
2
votes
0answers
123 views

Union of Sets in Locally Compact Hausdorff Space

Is it possible for an open set in a locally compact Hausdorff space to not be the union of an increasing sequence of compact sets? If so, given a regular Borel measure on such a space, how is it that ...
2
votes
0answers
192 views

Prove equivalent metric spaces

Let $X_1=[1,2]$ and $X_2=[0,1]$. Let $d_1$ denote Euclidean and let $d_2(x,y)=2|x-y|$ in $X_2$. Show that $(X_1,d_1)$ and $(X_2,d_2)$ are equivalent metric spaces. How do I do that?
2
votes
0answers
70 views

intro. to topology mendelson - closure in a subspace

I'm self studying intro to topology by Mendelson and I just completed a book problem and wanted to get input on whether it's okay. The problem statement is, Let $Y$ be a subspace of $X$ and ...
2
votes
0answers
142 views

Covering argument

In proving Harnak's inequality (I am referring to this article: "On Harnack’s Theorem for Elliptic Differential Equations"Communications on Pure and Applied Mathematics Volume 14, Issue 3 ), Moser ...
2
votes
0answers
146 views

Intersection form on manifolds with boundary

It is a "basic fact" that the intersection form of a closed oriented 4k-dimensional manifold is unimodular. (Could anyone point me to a reference to a proof of this fact?) What can be said about the ...
2
votes
0answers
380 views

If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact

Let $X$ be a Banach space and $C\subset X$. $\fbox{1}$ If $C$ is convex , weakly-closed and norm-bounded $\Longrightarrow$ $C$ is weakly-compact ? $\fbox{2}$ If $C$ is convex , weakly-closed ...
2
votes
0answers
53 views

Topological graphs

Given the universel covering space $\hat{X}$ of $X$ by $p:\hat{X}\rightarrow X$, there exists a bijection between subgroups $H<G=\pi_1(X,x_0)$ and covering spaces $\tilde{X}\rightarrow X$ with ...
2
votes
0answers
53 views

field lines terminating at infinity

A dipole consists of two equal and opposite point charges separated by a fixed distance. With two exceptions, all the electric field lines begin on one charge and end on the other. In the two ...
2
votes
0answers
61 views

A question on semi-stratifiable spaces

A space $(X,\tau)$ is called semi-stratifiable if there exists a function $g:\omega\times X\to\tau$ such that: (i) for any point $x$ of $X$ holds $\{x\}=\bigcap_{n\in\omega} g(n,x)$; (ii) for any ...
2
votes
0answers
52 views

Jordan curves, its interiors and the existence of a continuous function.

Let $L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$ is a Jordan curve, $O(t)$ is its interior and $H(t,s)=L_t(s)$. If $H$ is a homotopy from $L_0(s)$ to$L_1(s)$, is there exists a continuous ...
2
votes
0answers
156 views

a problem on metric spaces

I am reading the book by Burago and Ivanov "A course in metric geometry". I tried to do some problems but have some difficulties. For example, page 66 exercise 3.1.26: Let $(X, d)$ be a metric space ...
2
votes
0answers
101 views

Compute $df_1: ST_1^3 \rightarrow TSO(3)_I$

In short, the problem is to compute $df_1: T_1S^3 \rightarrow T_{I}SO(3)$, given $f: S^3 \rightarrow SO(3), r \in S^3, f(r) \in SO(3): f(r)(q) = rqr^{-1}, q\in R^3$. I just get to study differential ...
2
votes
0answers
129 views

Problems about continuity of $|f|$ and $f\vee g$; confusion about definitions

I can't seem to wrap my head around this notation of my textbook can some please explain to me what this says? What I am trying to show? (a) Given $f: D \to \mathbb {R}$, let $|f|$ be the ...
2
votes
0answers
88 views

Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
2
votes
0answers
108 views

Cantor set as a set of continued fractions?

Does the classical cantor set have a nice description as a set of continued fractions? I made a (superficial) search and didn’t find anything, but I’m very tired right now, so please forgive me that ...
2
votes
0answers
204 views

Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
2
votes
0answers
81 views

A question on star $\sigma$-compact spaces

A topological space $X$ is said to be star $\sigma$-compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a $\sigma$-compact subspace $K$ of $X$ such that $X = ...
2
votes
0answers
54 views

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable? Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space. ...
2
votes
0answers
38 views

continuous map question

Let $f: S^1\rightarrow \mathbb R$ be a continuous map. a) Let $A_1= \{(x,y) \in S^1\times S^1 |\ x\neq y, f(x)=f(y)\}$. b)$A_2=\{(x,y)\in S^1\times S^1| x=-y, f(x)=f(y)\}$ How many element does ...
2
votes
0answers
74 views

Very Simple Universal Covering problem

A space $X$ is constructed from two disjoint copies of $RP^3$ and a copy of the unit interval $I$ by gluing one end of $I$ to a point of one copy of $RP^3$, and gluing the other end of $I$ to the ...
2
votes
0answers
53 views

defining relationship between geometric entities (features)

I have different features located on a plane (2D); I want to define this structure mathematically in a way to represent their relations. Some of features are aligned in horizontally, vertically or ...
2
votes
0answers
274 views

Restriction of a covering map to a subspace

Let $p:X\rightarrow Y$ be a covering map and let $Y_0 \subset Y$. Show that $p|:p^{-1}(Y_0)\rightarrow Y_0$ is a covering map. Hint: Show first that if $V\subset Y$ is well-covered by $p$, then ...
2
votes
0answers
193 views

Example of a nontrivial finite covering map

A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
2
votes
0answers
155 views

Pullbacks as manifolds versus ones as topological spaces

Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd). Questions: Suppose that we ...
2
votes
0answers
213 views

Topological proof that this set is a topological manifold

let $S \subseteq \mathbb R^3 \times \mathbb R^3$ be the set of pairs $(x,y)$ where x,y are orthogonal unit vectors in $\mathbb R^3$. i am trying to show that this is a topological manifold without ...
2
votes
0answers
44 views

Topological inequivalent manifolds obtaining by removing a surface from a manifold

Are there any general techniques for classifying the inequivalent topologies that can be obtained by removing a 2-surface S from a 4-manifold M? I am particularly interest in the case where both M and ...
2
votes
0answers
76 views

Prove that you can't connect both pairs of opposite sides of a square without the two paths intersected.

Formally, let $$D=[-1,1;-1,1]\subset\mathbb{R}^2,$$ and let $f,g:[0,1]\to D$ be two continuous functions, such that $f(0)=(-1,0)$, $f(1)=(1,0)$, $g(0)=(0,-1)$, $g(1)=(0,1)$. Prove that ...
2
votes
0answers
24 views

How to build the space BTOP

Can anybody explain how is the procedure for building the space BTOP, which classifies microbundles of topological manifold ? Is there any good (and easy to read) references on this subject ? Thanks ...
2
votes
0answers
85 views

How to show that for any meager set $A$ in Baire Space, there is a nowhere dense set $C$, such that $A \subseteq C^*$?

Let $X$ be a subset of $\omega^{\omega}$, $X^{*}$ is defined as:$$\{y:(\exists x\in X,\exists N <\omega)(\forall n >N x(n)=y(n))\}$$ which consists of all sequences in $\omega^{\omega}$ that ...
2
votes
0answers
189 views

sub-basis for a topology on the real line

Consider the closed intervals $[a,b]$ as a sub-basis for a topology on the real line. Describe the resulting topology My attempt If $[a,b]$ is open, and $[a-1,a]$ is open, then the intersection of ...
2
votes
0answers
81 views

Γ-spaces and operads

I'm looking for a comprehensible reference that explains how $\Gamma$-spaces are related to $E_{\infty}$-operads. I've found some old publications but was hoping there are better references out there. ...
2
votes
0answers
63 views

Inverse limits and sums

Let $(X_\alpha)_{\alpha<\omega_1}$ be a family of compact metric spaces such that $X_\alpha$ is homeomorphic to a subspace of $X_\beta$ for $\alpha<\beta$. Can we regard the disjoint sum ...
2
votes
0answers
127 views

Quotient map from the $(2n+1)$-dimensional sphere into complex projective space is open.

We have a natural quotient map $$\phi\colon S^{2n+1}\rightarrow (\mathbb{C}^{n+1}\setminus{\{0\}})/\mathbb{C}^{*}=\mathbb{P}^n\mathbb{C}$$ and I want to see, that it's open. Denote by $$\iota\colon ...
2
votes
0answers
132 views

Homeomorphism between simply connected, closed 3 - manifold and 3-sphere.

The Poincare conjecture states that a simply-connected, closed 3-manifold is homeomorphic to the 3-sphere. Now that the conjecture has been settled, could someone tell me what this homeomorphism is ...
2
votes
0answers
275 views

topology-cofinite topolgy and co-countable topolgy

Let $T_1=\{U:X-U \text{ is finite for all of } X\}$. Then $T_1$ is the cofinite topolgy on $X$,where $X$ is an arbitrary infinite set. Then $T_1$ is not a Hausdorff space.Is it a regular space or a ...
2
votes
0answers
42 views

Understanding intuitively that any loop $p:[0,1] \to S^1$ is end point preserving homotopic to a loop which doesn't change direction.

I'm trying to understand intuitively the proposition that: Any loop $p:[0,1] \to S^1$ is end point preserving homotopic to a loop which doesn't change direction. Surely a loop round a circle ...
2
votes
0answers
272 views

Properly discontinuous action on a non-locally compact space

Let me begin with some definitions in order to avoid confusion. An action of a group $G$ on a space $X$ is proper if the map $G \times X \to X \times X$ given by $(g, x) \mapsto (x, gx)$ is proper, ...
2
votes
0answers
67 views

How to prove an isotopy relative to a point exist?

Let $M$ $ $ be a differential manifold, and $f$ a diffeomorphism on $M$ which is isotopic to $id$. Assuming that $x\in M$ is a fixed point of $f$ and the orbit of $x$ under the isotopy is a trivial ...
2
votes
0answers
147 views

Affine transformation

Let $S_1$ and $S_2$ be sets. Let $n_1$ be the cardinality of $S_1$ and $n_2$ be the cardinality of $S_2$. I assume that $n_1$ and $n_2$ are finite. Let $e$ be a function that maps members of $S_1$ and ...
2
votes
0answers
143 views

A Cartesian product of metric spaces is perfectly $\kappa$-normal

A space $X$ is called perfectly $\kappa$-normal if the closure of any open set (that is, every canonical closed set) is a zero-set. How can i prove this proposition directly? $Proposition$: A ...
2
votes
0answers
178 views

F-space and completely regular space

A completely regular space X is an F-space if for each functionally open set $M\subset X$ every continous function $f: M \rightarrow I $ continuously extendable over X. I want to prove that a ...
2
votes
0answers
19 views

Edwards-Anderson Hamiltonian of a Hopf link

I was calculating the Edwards-Anderson Hamiltonian of a Hopf link. A hopf link is like attachment 1. I have drawn the Seifert surface of that link. The surface is shown in attachment 2. It also ...
2
votes
0answers
114 views

Twisted tori: discrete and continuous

Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE: Motivation Let me introduce twisted (discrete) tori: Consider the Cartesian graph product $\mathcal{C}_n = C_n ...
2
votes
0answers
62 views

Continuity of linear form

Let $E=\mathbb{R}[X]$ We define $N:\, P \to \sum_{n=0}^{\infty} { |P^{(n)}(n)|}$ ($P^{(n)}$ being the $n$-th derivative) , it is not hard to prove that $N$ is a norm on $E$. Help me to study the ...
2
votes
0answers
193 views

Separable Banach Space

Let $X$ be a real separable Banach space. Let $A\subset X$ be the enumerable set, given by the separability. How can i define a continuous "bijective" function $f:A\rightarrow\mathbb{Q}$, where im ...
2
votes
0answers
62 views

Show that a node of a real curve is homeomorphic to a cross

This is related to this question. Let $C = \{(x,y) \in \mathbb{R};\; x^3 + x^2 - y^2 = 0\}$ equipped with the subspace topology of the euclidian plane. I want to show that there's a neigbourhood $U$ ...
2
votes
0answers
100 views

Separating points from open sets in a compact space without isolated points

Given that $S$ is compact and it has no isolated point. Show that given any nonempty open set $P$ of $S$ and any point $x\in S$, there exists a nonempty open set $V\subset P $ such that $x\notin \bar ...
2
votes
0answers
112 views

Form of weakly continuous linear functional

This was originally a problem in Stratila and Zsido's "Lectures on von Neumann algebras" (E.1.2). I've spent so much time working on it, and right now I cannot see how the result can be so simple. ...