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)

0
votes
2answers
79 views

Is $\mathbb{R}^2$ boundaryless?

I just have a quick question, as stated in the title. Is $\mathbb{R}^2$ boundary-less? Thank you very much. :-)
1
vote
0answers
36 views

If $f, g: S^1\to \mathbb C$ are two functions, what is a homotopy from $f=\frac{g}{\vert g\vert}$ to $g$?

If $f, g: S^1\to \mathbb C$ are two functions, what is a homotopy from $f=\frac{g}{\vert g\vert}$ to $g$? I just want to check whether my homotopy $H(x,t): (1-t)f+tg$ where $x \in S^1, t \in [0,1]$ ...
1
vote
1answer
58 views

how to determine the following set is countable or not?

How to determine whether or not these two sets are countable? The set A of all functions $f$ from $\mathbb{Z}_{+}$ to $\mathbb{Z}_{+}$. The set B of all functions $f$ from $\mathbb{Z}_{+}$ to ...
0
votes
2answers
56 views

Closure and pathwise connected

On $\mathbb R^2$ I consider the following set $G:=\{(x,\sin(\frac{1}{x}))| x\in (0,1]\}$ Well this set is pathwise connected (as a graph of a contionous function on an interval). The function is ...
2
votes
3answers
95 views

A problem on neighbourhood bases of a metric space

Let $(X,d)$ be a metric space and $ a \in X $. If $\{a\}$ is not open, then show that any neighbourhood base of $a$ is infinite.
0
votes
0answers
44 views

Cartwright-Littlewood Theorem

I have some question in the proof of the following theorem. I pose the question mark on each statment that i want to know. I would be so grateful if someone can help me. Thanks in advance An ...
1
vote
1answer
43 views

Do spaces satisfying the first axiom of countability have monotone decreasing bases for every point?

I'm facing some problems with this. Every proof I've read assumes this, although it is not obvious to me as of now. First axiom of countability as defined in my book- for every point $x\in X$, there ...
18
votes
1answer
169 views

Is there a “deep line” topological space in analogue to the “long line” $\omega_1\times[0,1)$?

I was reading about the "long line" $L=\omega_1\times[0,1)$ in the lexicographic order topology, which is locally like $\Bbb R$ except that it is "long" on one end, so there is no countable sequence ...
1
vote
1answer
94 views

Can a topological space satisfying the first axiom of countability, which is not Hausdorff but every sequence has a unique limit, exist?

My textbook (Principles of General Topology, by Pervin) says "A topological space $X$ satisfying the first axiom of countability is a Hausdorff space iff every convergent sequence has a unique limit." ...
3
votes
2answers
73 views

Is $\omega_1 ^\omega$ countably compact?

Give $\omega_1$ the order topology, and then $\omega_1 ^\omega$ the product topology. $\omega_1$ is countably compact, but what about this product? I attempted to prove it in two different ways, ...
4
votes
3answers
86 views

Compactness Proof

I have a doubt respect this theorem A compact subset $M$ of a metric space is closed and bounded. Proof by my lecture: For every $x\in \bar{M}$ there is a sequence $(x_n)$ in $M$ such that ...
1
vote
1answer
72 views

Extending characteristic function of a compact set while keeping relatively compact support

Let $X$ be a locally compact Hausdorff space, and $F\subset X$ compact. I want to extend $\chi_{F}$ to a non-negative continuous functional $f$ whose support is relatively compact (I'm working ...
3
votes
2answers
96 views

Can a set of non self-intersection points of a space-filling curve contain an arc?

Consider a continuous surjection $f:[0,1]\to[0,1]\times[0,1]$. It can be proved that set of self-intersection points must be dense. In the Hilbert curve, the set of self-intersections are points ...
1
vote
1answer
67 views

continous function and closed graph

$f:X \rightarrow \mathbb{R}$ is such that it's graph is closed where $X$ is compact . Can someone provide me some examples that it is not necessarily true that $f$ is continuous?
1
vote
2answers
74 views

How to show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$?

I really think I have no talents in topology. This is a part of a problem from Topology by Munkres: Show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$. I always have the feeling ...
1
vote
1answer
268 views

How to show $d(x,A)=0$ iff $x$ is in the closure of $A$? [duplicate]

This is a problem form Topology by Munkres: Let $X$ be a metric space with metric $d$ and $A$ is a nonempty subset of $X$. Show that $d(x,A)=0$ if and only if $x$ is in the closure of $A$. I ...
1
vote
1answer
263 views

Properties of a one to one continuous function from $[0, 1]$ onto itself

Let $f$ be a one to one continuous function from $[0, 1]$ onto itself. Show that (i) $f$ is a homeomorphism. (ii) $f$ is strictly monotone on $[0, 1]$ (iii) Is it true that if $f$ is strictly ...
1
vote
1answer
99 views

Is each space filling curve everywhere self-intersecting?

Consider a continuous surjection $f:[0,1]\to [0,1]\times[0,1]$. Is $$\{x:\exists(t_1\not=t_2) f(t_1)=f(t_2)=x\}=[0,1]\times[0,1]?$$
4
votes
1answer
87 views

Weak and strong topology on infinite dimensional spaces

Is there a simple example to show that the weak and strong topology on an infinite-dimensional space do not need to coincide? I have several ideas using differences in the weak and strong convergence ...
2
votes
1answer
93 views

Has $X\times X$ the following property?

Let $X$ be a topological space that satisfies the following condition at each point $x$: For every open set $U$ containing $x$, there exists an open set $V$ with compact boundary such that $x\in ...
5
votes
1answer
123 views

Definition simply connected in $\Bbb C$

I recently saw a different definition for simply connected which I had never seen before. A connected subset $\Omega\subset\Bbb C$ is called simply connected if the boundary is the image of a simple ...
8
votes
2answers
99 views

If $f:X\to Y$ is a one-to-one continuous mapping, and $Y$ is a $T_{2}$-space, then does $X$ too have to be a $T_{2}$-space?

Motivation: For any two points $y_{1},y_{2}\in Y$, there are disjoint open sets containing $y_{1}$ and $y_{2}$ separately, as $Y$ is a $T_{2}$-space. Say $f(x_{1})=y_{1}$ and $f(x_{2})=y_{2}$. Then, ...
1
vote
3answers
267 views

Prove that $f$ has a fixed point . [duplicate]

For $f:[a,b]\rightarrow [a,b]$ is a continuous . Prove that $f$ has a fixed point . Is that true if we change $[a,b]$ by $[a,b)$ or $(a,b)$.
0
votes
1answer
88 views

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable?

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable? What I've tried: I list these facts: 1 A space $X$ is a Moore space iff $X$ is a $\sigma$-space and a $p$-space. 2 If $X$ is a ...
2
votes
2answers
67 views

Compactifications of limit ordinals

I thought I knew that but it seems I don't. Let $\alpha$ be a countable, limit ordinal $\alpha>\omega$. Give $\alpha$ its order topology. What is the Stone-Čech compactification of $\alpha$? Is ...
0
votes
1answer
82 views

A possible contradiction to “A $T_{1}$ space is countably compact iff every countable family of closed sets has a nonempty intersection”?

My textbook says "A $T_{1}$-space is countably compact iff every countable family of closed sets having the finite intersection property has a non-empty intersection" (Principles of General Topology, ...
4
votes
1answer
61 views

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $A \subseteq \mathbb{R}$

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and $A \subseteq \mathbb{R}$ (i) If A is connected, is $f^ {−1} (A)$ so? (ii) If A is compact, is $f^{−1} (A)$ so? (iii) If A is finite, ...
2
votes
4answers
353 views

Finding a counterexample; quotient maps and subspaces

Let $X$ and $Y$ be two topological spaces and $p: X\to Y$ be a quotient map. If $A$ is a subspace of $X$, then the map $q:A\to p(A)$ obtained by restricting $p$ need not be a quotient map. Could you ...
1
vote
1answer
80 views

How to prove this fundamental relationship $ b=\ell+n-1$?

How to prove this fundamental relationship? In a network or circuit, number of loop, nodes and branches has to satisfy the following fundamental relationship: $$ b=\ell+n-1,$$ ...
1
vote
0answers
38 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
3answers
136 views

any open ball of radius $2$ is an infinite set?

Is it true that in an infinite metric space, any open ball of radius $2$ is an infinite set? for example $\mathbb{R}^2$ with discrete metric we have $d(x,y)=1\forall x\ne y$ so in this case also we ...
3
votes
4answers
83 views

Is the set $\{(x, y) : 3x^2 − 2y^ 2 + 3y = 1\}$ connected?

Is the set $\{(x, y)\in\mathbb{R}^2 : 3x^2 − 2y^ 2 + 3y = 1\}$ connected? I have checked that it is an hyperbola, hence disconnected am i right?
0
votes
1answer
100 views

$[0, ∞)$ onto the unit circle which is not a homeomorphism

Give an example of a continuous map of $[0, ∞)$ onto the unit circle which is not a homeomorphism. $f(x)=e^{ix}$ works?
1
vote
2answers
444 views

Proving derived sets are closed

I am following a proof of the statement The derived set(the set of accumulation points) $A'$ of an arbitrary subset $A$ of $\mathbb{R}^2$ is closed. in a book. It starts with Let $q$ be a ...
2
votes
2answers
69 views

Projections Open but not closed [duplicate]

I often read that projections are Open but generally not closed. Unfortunately I do not have a counterexample for not closed available. Does anybody of you guys have?
1
vote
2answers
76 views

morphisms on topological spaces

In the category of topological spaces: 1.) Show that a morphism is monic IFF it is injective 2.) Show that a morphism is epic IFF it is surjective 3.) Are there any morphisms that are monic and ...
3
votes
1answer
102 views

Tietze–Urysohn's lemma in $\mathbb{R}^n$

Let $F_1$ and $F_0$ be closed subsets in $\mathbb{R}^n$, $F_0\cap F_1=\varnothing$. How to build a $C^{\infty}$- function $f:\mathbb{R}^n\to \mathbb{R}$, such that $f|_{F_1}=1$, $f|_{F_0}=0$ and ...
2
votes
1answer
96 views

Proper map on from compact manifolds

I'm stuck on this statement. Could anyone please help me out? Let $X$ be a compact manifold, every map $f: X \longrightarrow Y$ is proper. The definition of proper: a smooth map between manifolds is ...
0
votes
1answer
31 views

How could we see that a pseudocomplete space is always pseudocompact?

A space $X$ is called pseudocomplete if it has a sequence $\{\mathcal B_n:n \in \omega\}$ of $\mathcal{\pi}$-bases such that for any family $\{B_n: n\in \omega\}$ with $B_n\in \mathcal B_n$ and ...
2
votes
2answers
54 views

Proving every open subset $H$ of the plane $\mathbb{R^2}$ is the union of open discs

I am going through the proof of Every open subset $H$ of the plane $\mathbb{R^2}$ is the union of open discs in a book (open disc is the standard open Euclidean disc or open ball). It goes ...
2
votes
2answers
104 views

Terminologies related to “compact?”

A set can be either open or closed, and there can either be a finite or infinite number of them. A "compact" set is one where every open cover has finite subcover. Is there such a thing as a set ...
2
votes
1answer
262 views

Do all the open sets containing a limit point of an infinite countably compact subset have to contain infinite points?

Say an infinite set is countably compact (if set $E$ is an infinite countably compact set, it contains at least one limit point within itself). Let $x$ be one such limit point of $E$. My textbook says ...
2
votes
0answers
47 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 ...
7
votes
2answers
158 views

Complement of a Topology

Let $(X, \tau)$ topology, I was wondering, if given $$\tau'=\{A^C \mid A \in \tau\}$$ did $\tau'$ also a topology on $X$? If so, why? Thank you.
3
votes
1answer
167 views

Prove that a standard torus is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$

I was asked to prove that a standard torus(which means we don't consider those pathological cases where it intersects with itself, e.g horn torus) is diffeomorphic to $ \mathbb S^1\times \mathbb S^1$. ...
0
votes
2answers
81 views

Bounded partial quotients set is nowhere dense

I've stumbled upon a claim that the set: $$ B_N = \{[a_0;a_1,a_2,...] | \exists n_0 >0\forall n\geq n_0 a_n<N\} $$ for some $N$, is nowhere dense (and closed). Unfortunately, I have found that ...
4
votes
0answers
110 views

Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and ...
3
votes
3answers
172 views

Homomorphism/map in both direction implies isomorphism/homeomorphism or not?

I was working on a homework, and my first attempt get me to a deadend, but I was eventually able to solve it using a different method. But the fail attempt make me curious, and I wonder if it could ...
2
votes
1answer
129 views

Notation for set of all closed sets

Is there a common notation for the set of all closed sets of a topological space? I have been using $(X,\tau)$ to denote a topological space with $\tau$ being the topology, set of all open sets. I ...
1
vote
2answers
62 views

Applying the contra positive of the finite intersection property

I'm reading a proof which has the following setting. I have a family $D$ of compact sets with empty intersection. The next line takes a finite subset of $D$ with empty intersection. This is ...