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
53 views

What's the most general geometry branch?

What is the most general geometry of curves and surfaces? For example, at curves, we define in differential geometry the tangent vector as the derivative of a regular curve, but visually many other ...
0
votes
1answer
33 views

$\mathbb{R}^{2}$ and $\mathbb{R} \times [0, +\infty]$ are homotopy equivalent, but not homeomorphic

So, let's consider $M=\mathbb{R}^{2}$ and $N= \mathbb{R} \times [0, +\infty]$ - two topological spaces. Since $\pi_{1}(M)=\pi_{1}(\mathbb{R}) \times \pi_{1} (\mathbb{R}) = \{0 \}$ (since $\mathbb{R}$ ...
0
votes
0answers
25 views

Does total order imply linearisation

Suppose $X$ is a totally ordered set. Does this mean that $X$ can always be linearised? I mean can $X$ be always written in a linear order like $\mathbb{R}$ ? I came across this question when I was ...
0
votes
2answers
39 views

Group homomorphism on unit circle

For $n\in \mathbb{Z}$, define the map $f_n:S^1\to S^1$ as $f_n(z)= z^n$, where the unit circle $S^1$ is observed as the subspace $\{z\in\mathbb{C}|\ |z|=1\}$. How would one compute the induced group ...
1
vote
2answers
25 views

Extreme value theorem, without Heine Borel.

I was wondering, if there are any mistakes, in this proof of the extreme value theorem: Theorem. Let $X$ be a compact set and $f:X\rightarrow\mathbb{R}$, s.t. $f$ is continuous. Then there exists ...
4
votes
1answer
51 views

The union of a sequence of closed sets with empty interiors has empty interior in a compact Hausdorff space?

This is problem 5 in section 27 of Munkres' TOPOLOGY, 2nd ed Let $X$ be a compact Hausdorff space; let $\{A_n\}_{n\in \mathbb{N}}$ be a countable collection of closed sets of $X$. If each set $A_n$ ...
1
vote
1answer
48 views

Does proper map $f$ take discrete sets to discrete sets?

Suppose $f:X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Are the following results true? $1$. The map $f$ takes discrete sets to discrete sets. $2$. If $f$ is ...
1
vote
1answer
48 views

Prove that the convergent sum of a real sequence is a metric

I want to show that $$ \varrho(\{a_n\},\{b_n\})=\left(\sum_{n=0}^\infty{(a_n-b_n)^2}\right)^{1/2} $$ is a metric, where $\{a_n\}_{n\in\Bbb N}\in \ell_2$, and $\ell_2$ is the set of all real sequences ...
4
votes
1answer
66 views

Questions about a topological proof of the FTA

I'm a high school student, curious about proofs of the Fundamental Theorem of Algebra. Specifically, I've been thinking about one of the topological proofs of the theorem, given in Courant's book, ...
0
votes
1answer
31 views

Non-empty intersection of specific sets

For any set Y (to begin with, it may be countable), given a collection of relations $$R = \{R_y \subseteq \{0,1\}^Y \mid y \in Y\},$$ having the finite intersection property and such that for ...
1
vote
1answer
25 views

Every nontrivial linear functional is open

Let $X$ be a normed linear space and let $f:X\to \mathbb K$ be a nontrivial linear functional. I want to prove that $f$ is open. I tried as follows: Let $E$ be an open set in $X$ and let $y\in f(E)$. ...
0
votes
1answer
46 views

does the closure of interior of a set equal to closure of this set?

Does the $\text{Cl}(\text{Int} A)=\text{Cl}(A)$? Here "Cl" denotes closure, "Int" denotes interior. That is not a duplicate of the question of "does the closure of interior of a set equal t the ...
2
votes
3answers
55 views

Does there exist a surjective continuous map $D^2 \to S^1$?

By considering the induced homomorphism on the fundamental groups, we know that there is no retract $D^2 \to S^1$. But is there any continuous surjection from $D^2$ to its boundary? It seems unlikely ...
2
votes
2answers
50 views

Topological proof that the interval $[a,b)\subset \mathbb{R}$ is not closed

I want to prove that the interval $[a,b)\subset \mathbb{R}$ is not closed using the definition that a set $A$ in a topological space $X$ is closed iff its complement $X-A$ is open. Here, the topology ...
3
votes
1answer
55 views

In $\mathbb Q_p$, proving every open ball is the disjoint union of more than one open ball

I'm reading the Foundations chapter of Gouvea's p-adic Numbers: An Introduction, and I'm trying to solve the following problem he poses to the reader: Take the $p-$adic absolute value on $\mathbb ...
3
votes
1answer
39 views

Remove one ring of Borromean rings in 3-sphere: linked or unlinked?

We know Borromean rings in a 3-sphere $S^3$ can be unlinked if we remove one of the three rings. Here let us consider a slight different procedure. If we remove the neighbored solid torus $B^2 \times ...
-1
votes
0answers
38 views

Homeomorphism from unit ball to unit sphere [on hold]

Consider the unit sphere in $\Bbb R^3$ given by $\{(x,y,z) \in \Bbb R^3|x^2+y^2+z^2 =1\}$. Let $p$ be a point in this unit sphere. Question: How can I construct an open set U within this unit sphere ...
-5
votes
2answers
44 views

proving f is continuous iff it takes limits to limits [closed]

How to show that iff $x_i\to x$ implies $f(x_i)\to f(x)$ then f is continous? For metric Spaces. Continuit definition the standard epsilon delta
0
votes
0answers
19 views

Mollifiers and smooth path connetction

Generaly it is about smooth path connection. I have a two smooth paths f,g and a points x,y,z of open subset of $R^n$ such that f is smooth path from a to y and g is smooth path from y to z. I define ...
0
votes
1answer
18 views

Can a point $z$ which belongs to a closed set be a limit point of an open set which is disjoint from the closed set in topological space $X$?

Say $X$ be a topological space, and $U$ and $V$ are open and closed sets respectively. Furthermore, $U$ and $V$ are disjoint. Now there is a point $z \in V$. Is it possible for the point $z$ to be a ...
0
votes
2answers
67 views
+50

Point-set topology

I am about to begin a self-study project in point-set topology. I am a final year undergraduate. I am looking for suitable resources, I have so far come across Munkres' textbook and would like to find ...
0
votes
1answer
38 views

Prob. 3 (b), Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How does the $K$-topology on $\mathbb{R}$ differ from the usual topology?

Let $$ K \colon= \left\{\ \frac{1}{n} \ \colon \ n \in \mathbb{N} \ \right\},$$ and let the $K$-topology on $\mathbb{R}$ be the one having as basis all open intervals $(a,b)$ and all sets of the form ...
0
votes
1answer
35 views

Showing the attractor of an IFS is either connected or totally disconnected

I came across this execise in a problem set about Iterated Function System (IFS) and fractals: "Show that the attractor of an IFS of the form $\{\mathbb{R};~ax+b, cx+d\}$ where $a,b,c,d \in ...
0
votes
0answers
26 views

The Coproduct of two spaces is the same as the disjoint union and is homeomorphic to the union when the spaces are disjoint

Let $X$ and $Y$ be two topological spaces. Consider the set $$X \cup Y = \{ x\; |\; x \in X \text{ or } x \in Y\}$$ In all the following I suppose that $X$ and $X$ are disjoint. I want to ...
0
votes
3answers
79 views

What are some simple examples illustrating the definition of “cover”

In my class the word "cover" is used very informally such as this set covers another set (this is for a class in PDE not topology by the way). Can someone provide a trivial example of cover to get ...
5
votes
8answers
494 views

What is the mathematical distinction between closed and open sets?

If you wanted me to spell out the difference between closed and open sets, the best I could do is to draw you a circle one with dotted circumference the other with continuous circumference. Or I would ...
5
votes
0answers
82 views

Hatcher 2.1.10…

Hatcher asked a question in chapter $2$ (a) Show the quotient space of a finite collection of disjoint $2$-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic ...
0
votes
2answers
35 views

A base generates an unique topology?

I was confused by this. Let $X$ be {$a,b,c$}, Let $\mathcal{B}$ be {{$a$},{$b$},{$c$}}. Let $ \mathcal{T}$ be {$X, \emptyset$, {$a$}, {$b$}, {$a,b$}}. Let $ \mathcal{T'}$ be {$X, \emptyset$, ...
1
vote
0answers
17 views

Disjoint Union of Completely Regular Spaces

I am trying a new approach to an already-solved problem, but I need help to see if I'm on point. Munkres Chapter 53, question 6 [abridged] asks, given a covering map $p: E \to B$: Show that "if $B$ ...
-1
votes
2answers
65 views

Decomposing 2-sphere into two homeomorphic subspaces [closed]

Can a 2-dimensional sphere be decomposed into two disjoint homeomorphic subspaces? If yes, can these subspaces be non-discrete / connected / have some other good properties?
0
votes
0answers
29 views

Construction of a Radon measure from a certain family of compact subsets

Let $X$ be a locally compact Hausdroff space. Let $\Gamma$ be a family of compact subsets of $X$ with the following properties. 1) $\emptyset \in \Gamma$. 2) $K\cup L \in \Gamma$ whenever $K ...
1
vote
1answer
43 views

simplicial homology definition

I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105: Hatcher ...
1
vote
1answer
28 views

Limit vs interior definition of continuity

Suppose I have two topological spaces $X$ and $Y$ whose topologies are defined by interior operators $\text{int}_X$ and $\text{int}_Y$ respectively, as well as a function $f$ with domain $I$ (for ...
0
votes
1answer
25 views

Boundary preserving map

Let $K\subseteq\mathbb{R}^2$ be a compact set. Is it true that for a continuous map $p:K\to\mathbb{R}^2$ we have: $p(\partial K)=\partial p(K)$? Are there any generalizations? P.S. Note that ...
0
votes
0answers
31 views

Pro-completion of finite algebras as Stone algebras

Recall that a profinite algebra (e.g. group, monoid, or whatsoever) is a cofiltered/inverse limit of finite algebra. In Johnstone's Stone space, he showed that finite discrete algebras are finitely ...
1
vote
1answer
34 views

Show that if $(X,d)$ is compact then, every open covering of $X$ has a Lebesgue number.

Let $(U_i)_{i \in I}$ be an open cover of a metric space $(X,d)$, a number $\epsilon >0$ is called a Lebesgue number of $(U_i)_{i \in I}$ if for all $x \in X$ exist $j \in I$ such that ...
3
votes
2answers
48 views

An infinite dimensional normed linear space is the union of two disjoint convex sets

Let $X$ be an infinite dimensional normed linear space. I want to show that there exist two disjoint convex sets $C_1$ and $C_2$ such that $X=C_1\cup C_2$ and both $C_1$ and $C_2$ are dense in $X$. I ...
0
votes
1answer
27 views

Continuous function between two topological spaces: an ELEMENTARY property. [duplicate]

I'm reading the first chapter of a book on general topology. It has a lot of small, simple exercises on almost all pages and I try to do them all to fully understand the subject. This one I did not ...
3
votes
0answers
39 views

If a set is Hausdorff relative to one topology, can it be compact relative to a strictly finer topology?

Let $\tau_1$ and $\tau_2$ be two topologies on set $X\neq\phi$ such that $(X, \tau_1)$ is Hausdorff and $\tau_1 \subsetneq \tau_2$. Can $(X, \tau_2)$ be compact? My effort: Suppose that $(X, ...
1
vote
2answers
64 views

Is this condition on continuity extraneous or troublesome?

I was trying to motivate the use of open sets for defining continuity (as in topology or metric spaces). I came to formulate the following definition of continuity for a function $f: X \rightarrow ...
-4
votes
0answers
21 views

Metric spaces and compactness [closed]

Let $X$ be a metric space. If for all compact $K$, the set $K\cap F $ is closed, then $F$ is closed.
1
vote
3answers
35 views

Statment concerning open sets and closures

I found the following line in a proof (from a good book) concerning locally compact spaces: Since $A$ and $B$ are both open and $A \cap B = \varnothing$, it follows that $\bar{A} \cap B = ...
12
votes
2answers
198 views
+200

Maximum $C$ such that every shape in $\Bbb R^2$ with area $<C$ can be placed to avoid $\Bbb Z^2$

For $C=1$, it has been proved here that every shape in the plane having area less than $1$ can be translated and rotated so that it does not touch any element of $\mathbb Z^2$. (In fact, for $C=1$, ...
3
votes
4answers
286 views

Little confusion about connectedness

Consider $X=\{(x,\sin(1/x)):0<x<1\}$. Then clearly $X$ is connected , as it is a continuous image of the connected set $(0,1)$. So, $\overline X$ is also connected , as closure of connected set ...
11
votes
1answer
132 views

Is it possible to develop differential geometry without points?

I read about pointless topology and locale theory, and become curious about this topic. For example, there is the concept "differential manifold" corresponds to "topological manifold". As this, are ...
4
votes
1answer
22 views

How to make a topology out of $N$ that involves convergent / divergent sets.

Let $N$ be the naturals $1, 2, \dots$ Call a subset $A$ of $N$ convergent if the reciprocal sum $\sum_{a \in A} \frac{1}{a}$ converges. Similarly call as set divergent if the sum diverges. Notice ...
-3
votes
2answers
38 views

Three questions from σ-compact spaces and topological groups [on hold]

every locally compact subgroup of a Hausdorff group is closed. A Hausdorff and $σ-$compact space X is a Baire space if and only if the set of points at which is $X$ is locally compact is dense in ...
-2
votes
1answer
33 views

A locally compact Hausdorff space which is not a metric space [on hold]

I need three simple examples. A locally compact Hausdorff space which is not a metric space. A locally compact topological group which is not Hausdorff. (my definition of topological group does not ...
2
votes
1answer
40 views

Haar measure on locally sigma-compact metric groups

Haar measure on locally sigma-compact metric groups $G$ is a metric group, if $G$ is a topological group meanwhile $G$ is a metric space(compatible with topology). We know that there exist a Haar ...
2
votes
0answers
39 views

On the properties of an interesting set on the real line…

Let $K$ be the set of all real numbers of the decimal form $$ 0.\;e_1\;\underbrace{0}_{1!\text{ times}}\;e_2\;\underbrace{00}_{2!\text{ times}}\;e_3\;\underbrace{000000}_{3!\text{ ...