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)

3
votes
0answers
54 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
0
votes
0answers
16 views

Family of functions depending continuously on a parameter space WRT the $L^1$ norm

The material I'm reading involves a family of functions induced by a parameter space homeomorphic to an open disk. It attempts to show that the functions depend continuously on this parameter with ...
0
votes
1answer
79 views

Connectedness and compactness of K-topology

Let $T_K$ be the K-topology on $\mathbb{R}$, this is, the topology generated by the collection of all open intervals $(a,b)$ and the sets of the form $(a,b)-K$, with $K=\{1/n, n \in \mathbb{Z}^+\}$. ...
7
votes
1answer
126 views

Compact subspace of a covering space

I've been working through Massey's A Basic Course in Algebraic Topology and I've gotten stuck on the following exercise (V.8.4): Let $X$ be a regular topological space, and $(\tilde{X}, p)$ a ...
3
votes
0answers
35 views

What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
1
vote
0answers
62 views

Is $f^n_b$ a surjection?

For the purposes of this question, let $\mathbb{N}_k = \mathbb{N}_{\geq k} = \left\{n \in \mathbb{Z} | n \geq k \right\}$ and $\mathbb{R}_+ = \mathbb{R}_{\geq 0} = \left\{x \in \mathbb{R} | x \geq 0 ...
1
vote
1answer
95 views

Number of subsets/open subsets/closed subsets of a metric space.

Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces which have the same infinite set $X$, but the different metrics $d_1$ and $d_2$. Denote the collection of subsets $X$ by $S$, and the collection of ...
2
votes
1answer
33 views

Proving the Weierstrass M-Test with topology

I've encountered some theorems in analysis that are ultimately provable in a more elegant way with topology. So, is there a topological proof of the Weierstrass M-Test, ideally not using terribly ...
0
votes
1answer
65 views

Continuity, and continuity in topology.

Metric spaces: Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r) Definition of open set: "A subset $O$ of ...
0
votes
1answer
10 views

Need help understanding proof relating to continuous functions from compact spaces

In the following proof on page 174 of Munkres, I don't understand why the following statement is true: If $A$ has no largest element, then the collection $\{(-\infty, a): a \in A\}$ forms an open ...
3
votes
3answers
78 views

Determine whether Set is closed,whether it is open, whether it is bounded and whether it is compact

QUESTION: $f(x,y)=y-\displaystyle\frac{1}{x^2}$ Consider the set $S = \{(x,y) \in D: x > 0,\ f(x,y) > 0\}$, Where $D$ is the domain of $f$. Sketch the set $S$ in the plane. Determine whether ...
1
vote
1answer
27 views

Subbase of a topology containing prime ideals (commutative ring)

Let $A$ be a commutative ring. Prove that the set of the ideal primes of $A$, along with $A$, is a subbase of some topology on (the subjacent set of) $A$ and that the complements of the prime ideals ...
1
vote
1answer
41 views

Countably local finiteness and a related(?) property

I'd like to know how the following properties are related: 1.) $\mathcal{O}$ is a cover of $X$ such that every point of $X$ has a neighborhood that intersects at most countably many members of ...
1
vote
1answer
24 views

Nets and directed sets problem

I am trying to solve the following exercise: Let $\Lambda$ be a directed set, and for each $\alpha \in \Lambda$ let $\Gamma_\alpha$ be a directed set. Suppose that for each $\alpha \in \Lambda$ there ...
2
votes
2answers
38 views

Explanation of a theorem of the topology generated by a subbasis

In my one of lecture notes, it says that the Topology $\tau_S$ on a set $X$ generated by a subbasis $S$ is $\tau_S=\{u\subset X \mid u\text{ is the union of finite intersections of elements of }S\}$. ...
2
votes
2answers
132 views

invertible matrices connected or not

The question asks "Is the set of all 3 by 3 real invertible matrices connected or not?" My intuitive idea is that we can establish a separation consisting of matrices with positive and negative ...
1
vote
1answer
51 views

A continous map between the two torus and the torus

Let $\Sigma$ be the doubled torus (a compact oriented) surface of genus 2) and let $T$ be the torus. Suppose $f: \Sigma \rightarrow T$. Prove that $f$ is not a local homeomorphism. Attempt at ...
0
votes
0answers
31 views

A Local Homeomorphism Between Compact Connected Hausdorff Topological Spaces

Prove that a local homeomorphism between compact, connected, Hausdorff spaces is a covering map of finite degree. Attempt at solution: Let $f:M\rightarrow N$ be the local homeomorphism. Since $N$ is ...
1
vote
2answers
33 views

Show that $K:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ is a topological group.

As the title already says I have to show that $$ K:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\} $$ is a topological group. First of all, $K$ is a group concerning the ...
1
vote
2answers
29 views

continuity on co-countable topology

Please help me.. I am looking for function like this $$f:\mathbb{R}_\text{co-count}\to \mathbb{R}$$ which is continuous but is not a constant function. Can any one give me an example?
5
votes
0answers
124 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
1
vote
1answer
14 views

How to prove this $\vert C^*(X)\vert \leq 2^{d(X)}$

The continuous functions is determined by a dense subset of X, that is, $\vert C^*(X)\vert\leq\vert C^*(D)\vert$. The density $d(X)$ of a space $X$ is the smallest cardinality of a dense subset of ...
2
votes
1answer
29 views

On nets convergence

I am trying to prove the following statement If $(x_{\alpha})_{\alpha \in \Lambda}$ verifies that every sub-net has a sub-sub-net which converges to $x$, then $(x_{\alpha})_{\alpha \in \Gamma}$ ...
1
vote
2answers
45 views

Show that if $x$ is a limit point of $A\subset X$ and $f:X \rightarrow Y$ is continuous then $f(x)$ is a limit point of $f(A)$

Suppose $f:X\rightarrow Y$ is a continuous function between the topological spaces $X$ and $Y$. Suppose that: $A \subset X$ and that $x$ is a limit point of $A$. Show that $f(x)$ is a limit point of ...
4
votes
0answers
92 views

Surgery presentations and gluing

It is well known that every closed oriented 3dimensional manifold can be obtained from a framed link in $S^3$. Let $M$ and $N$ be topological 3-manifolds with boundary such that boundaries $\partial ...
2
votes
3answers
79 views

What subsets $A$ of $\mathbb{R}^2$ are such that $\partial(A)=\partial(\partial(A))$?

What subsets $A$ of $\mathbb{R}^2$ are such that $\partial(A)=\partial(\partial(A))$? Is it necessary for a set to have an empty boundary for this property to hold?
3
votes
2answers
58 views

Intuition for the compactness of real projective space $\mathbb{R}\mathbb{P}^n$.

I want to have an intuition for why the $n$-dimensional real projective space defined as $$\mathbb{R}\mathbb{P}^n:=\mbox{set of 1-dimensional subspaces of }\mathbb{R}^{n+1}$$ is compact. I don't see ...
2
votes
2answers
31 views

Cocompact group actions have cobounded orbits

Assume $X$ is a complete, locally compact, geodesic metric space (in particular, $X$ has closed balls are compact, by the Hopf-Rinow Theorem). Assume $G$ acts isometrically on $X$. We say $Q\subset X$ ...
1
vote
0answers
27 views

Seperating neighborhoods of infinite sets in normal topological spaces

Let $(T,\tau)$ be a normal topological space, let $(x_n)_{n\in \mathbb{N}}\subset T$ be a discrete subset. Are there disjoint neighborhoods $U_n(x_n)\in\tau$?
0
votes
1answer
59 views

Is there a name for continuous functions $\Omega \rightarrow \mathbb{R}$ that can be continuously extended to $\overline{\Omega}$?

Given topological spaces $X$ and $Y$ together with a subset $\Omega \subseteq X$, is there a name for those continuous functions $f : \Omega \rightarrow Y$ such that $f$ can be extended to a ...
2
votes
1answer
30 views

Show that a finite union of compact subspaces of a topological space $X$ is compact.

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify my proof or offer suggestions for improvement? Show that a finite ...
2
votes
1answer
101 views

Modern mathematics for dummies

I have poor university mathematical education, but Math fascinates me, so I decided to educate myself for a bit. I know there is a dozen modern mathematical fields I know nearly nothing about like ...
1
vote
0answers
31 views

Prove that every subset of $\mathbb{R}$ is compact in the finite complement topology.

I need help with my proof in particular. I am aware that there is a similar question elsewhere. Can someone please verify my proof or offer suggestions for improvement? Prove that every subset of ...
1
vote
3answers
70 views

Prove $f: X \rightarrow Y$ is continuous if $A_a$ is closed and $f|A_a$ is continuous for any $a$ and $\text{{$A_a$}}$ is locally finite collection

Let $f:\bigcup_{\alpha}A_{\alpha} \rightarrow Y$ be a function between the topological spaces Y and $X=\bigcup_{\alpha}A_{\alpha}$. Suppose that $f|A_{\alpha}$ is a continuous function for every ...
1
vote
1answer
35 views

Closure operator and topology problem

Statement If $c:\mathcal P(X) \to \mathcal P(X)$ is a closure operator on $X$, then the set $\tau=\{U \in \mathcal P(X) : c(X \setminus U)=X \setminus U\}$ is a topology on $X$. First let me write ...
0
votes
1answer
24 views

Neighbourhood filter system exercise

Problem Let $X$ be a set. A neighbourhood filter system $\mathcal F$ on $X$ is a rule that assigns to each element $x \in X$ a family $\mathcal F_x \subset \mathcal P(X)$ such that (1) if $x \in X$, ...
0
votes
1answer
68 views

Sets identities on topological space [duplicate]

I am trying to show the following identities Suppose $X$ is a topological space and let $A \subset X$, then: $(a) int(X \setminus A)=X \setminus \overline{A}$ $(b) \overline{X \setminus A}=X ...
1
vote
0answers
31 views

Computing the tangential and cross components of one quantity using gnomonic projection

I have a spin-2 field given called shape distortion of galaxies as $$\gamma=\gamma_1+i\gamma_2=|\gamma|e^{-2i\phi}$$ where $\phi$ is the orientation angle. If this quantity has been measured on ...
3
votes
3answers
170 views

In a non-Hausdorff space, can a compact subset fail to be closed?

In a Hausdorff space $X$, every compact subset $Y$ is closed. So if I relax the condition on $X$ being Hausdorff, is it possible compact subset $Y$ of $X$ not being closed?
0
votes
0answers
26 views

Homotopic maps of a compact polyhedron

My friend and I are trying to solve the following exercise. Problem: Let $X \subset \mathbb{R}^n$ be a compact polyhedron. Show that there exists $\alpha > 0$ such that for any pair of maps $f, g ...
1
vote
0answers
23 views

Question about holomorphic proper maps

Let $U, V$ be connected open subsets of $\mathbb{C}$ and $f: U \to V$ which is holomorphic and proper. I am trying to show that $f$ is onto. Here is my attempt at a proof. Let $A = \{v \in V: ...
1
vote
1answer
36 views

$X \models S_f(\mathcal O,\mathcal O)$ implies $X \models S_1(\mathcal O,\mathcal O)$

I have a problem with a begining of a proof in a course that I am reading. I bring here the definitions and statement: Definition (Rothberger): For $k \in \mathbb N$, a space $\langle X,O \rangle$ ...
0
votes
3answers
37 views

Closed set between two open sets

If $U$ and $V$ are open subsets of $\mathbb{R}^{d}$ such that $U \subseteq V$, does there exist a closed set $E$ such that $U \subset E \subset V$?
0
votes
0answers
30 views

Can we call the boundary of a subset of a topological space “partial X”?

Intuitively, one might be tempted to say $\partial S$ (the boundary of $S\subseteq X$ for X a topological space) as "partial X". Is this formally valid?
0
votes
0answers
21 views

Paracompactness and partitions of unity

For Hausdorff spaces, paracompactness is equivalent to finding subordinate partitions of unity for any open cover. I am confused about the "easy" step. If $f_j$, $j \in J$ is a partition of unity ...
0
votes
1answer
58 views

Is there a name for this variant on “continuous function”?

Let $X$ and $Y$ denote topological spaces. Then a function $f : X \rightarrow Y$ is said to be continuous iff for all $U \in \mathcal{P}(Y)$, it holds that if $U$ is open in $Y$, then $f^{-1}(U)$ is ...
0
votes
2answers
34 views

Continuous functions on compact Hausdorff space.

There is a well known theorem that says that if $X$ is a compact Hausdorff space, then the space $C(X)$ of the continuous functions on $X$ is a complete Banach space with the sup norm. It's clear why ...
0
votes
0answers
24 views

Show that $\cup A_n$ is connected.

Can someone please verify my proof or offer suggestions for improvement? Let $\{A_n\}$ be a sequence of connected subspaces of $X$, such that $A_n \cap A_{n+1} \neq \varnothing$ for all $n$. Show ...
0
votes
0answers
27 views

Doubt about local flatness of low dimensional embeddings

I would like to know if it is possible to have a simple curve $\gamma $ on a surface $S$ such that $\gamma$ is compact and embedded (i.e. with respect to the topology induced from $S$ it is ...
4
votes
1answer
51 views

Does the converse of Tychonoff's theorem hinge on the axiom of choice?

Tychonoff's theorem:$\phantom{---}$ If $A$ is a non-empty index set and $X_{\alpha}$ is a non-empty compact topological space for every $\alpha\in A$, then $X\equiv\times_{\alpha\in A} X_{\alpha}$ is ...