Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Filter by
Sorted by
Tagged with
0
votes
0answers
11 views

Equivalence (Or non-equivalence) of definitions for Metric Spaces

So, I've stumbled across a definition of a metric space that has given me some pause. The definition of a Metric Space that I've always used has been the following: A Metric Space is a set $X$ ...
0
votes
0answers
13 views

Image of compacts sets under a pseudo-continuous functions is bounded

I'm reading a paper where the author makes the following remark: If $K\subseteq 2^\omega$ is compact and $\phi:K\rightarrow\omega^\omega$ is pseudo-continuous, then $\phi''K$ is bounded. Let me put ...
0
votes
0answers
18 views

$h_*:S_*(X\times Y)\to S_*(X\times Y)$ a homomorphism of natural chain complexes. If $h_0=Id$, show that $h_*$ is homotopic to $Id$

Let $X$ and $Y$ be topological spaces and $h_*:S_*(X\times Y)\to S_*(X\times Y)$ a homomorphism of natural chain complexes. If $h_0=Id$, show that $h_*$ is homotopic to $Id$. To solve this problem I ...
2
votes
2answers
26 views

Is any open set of $(0,\infty)\times S^{k-1}$ a countable disjoint union of products $I\times A$, where $I$ is an interval and $A$ is open?

Let $V$ be an open set in $(0,\infty)\times S^{k-1}$. I want to prove that $V$ can be written as a countable disjoint union of the form $$V=\coprod_{n=1}^\infty I_n\times A_n,$$ where each $I_n$ is an ...
-2
votes
1answer
26 views

Application of hyperspheres

Recently I've been studying the the volume of an n-ball. Do hyperspheres (or their volume/surface formulas) have any real-world applications?
-1
votes
1answer
13 views

Topology of derived sets and subset [on hold]

if A is a subset of C ,then A' is a subset of C' A' denotes set of all limits points of A
0
votes
0answers
33 views

A product of two connected spaces is connected.

I know the standard proof of this theorem, but is there any problem with this proof: Let $\varnothing \neq E \subseteq X \times Y$ then $E=\bigcup U_\alpha \times V_\alpha$ where $U_\alpha, V_\alpha$ ...
2
votes
1answer
33 views

Space of Bounded Functions

Let $B(\mathbb{R},\mathbb{R})$ be the space of all bounded functions on $\mathbb{R}$. Is it possible to define two norms on this space generating two different topologies? The only norm I know on ...
0
votes
1answer
22 views

Index of a point on a Jordan curve

Let $p$ be a point on a Jordan curve $J$. Let $\gamma$ be a loop in the bounded component of $\mathbb{C} \backslash J$. Show that $ind(\gamma,p) = 0$. Can we just tell that the curve is contractible ...
0
votes
0answers
17 views

Is it possible to make a stereographic projection of a stereographic projection?

Since we can't visualize a 3-sphere directily, I was wondering how does its stereographic projection look like, and found this neat demonstration: https://demonstrations.wolfram.com/...
1
vote
1answer
23 views

Regarding Urysohn's metrization theorem proof

At Munkres' topology, he showed two different proof of Urysohn's metrization theorem. In the first version of the proof, he tried to construct some continuous function $F$ from a given space $X$ ...
0
votes
0answers
20 views

A question about non-wandering points and their definition

As I read in a paper the definition of non-wandering points is this : $ x \in X $ is a non-wandering point if for every neighborhood $U$ of $x$ there exists a natural number $n$ suchthat : \...
1
vote
1answer
47 views

Closed in $\mathbb{R}^{2}$ vs closed in $\mathbb{R}$ [on hold]

Define the following subsets of $\mathbb{R}$ and $\mathbb{R}^{2}$ respectively: $A=\left\{\frac{1}{n}: n \in \mathbb{N}\right\}$ $B=\left\{(x,\frac{1}{x}): x\in\mathbb{R} \smallsetminus \left\{0\...
16
votes
3answers
2k views

Trying to visualize the hierarchy of mathematical spaces

I was inspired by this flowchart of mathematical sets and wanted to try and visualize it, since I internalize math best in that way. This is what I've come up with so far: Version 1 (old diagram) ...
-1
votes
1answer
66 views

Proving that a set is an open set by using the concept of accumulation points

I need help to prove that, by using the definition of closed sets and accumulation points, the following characterization of open sets holds. Definition 1. An accumulation point of $S \subseteq R$ is ...
0
votes
1answer
36 views

Is $(0,1)\cup[2,3)$ not Dedekind complete or does it have a gap somewhere?

I learned that a linear order is connected if and only if it is Dedekind complete (any nonempty bounded subset has lub and glb) and has no gap ($\forall a < b \in L, (a, b) \neq \emptyset$). I ...
2
votes
1answer
55 views

Image of a Topologically Regular Set?

This question has come up as a small question in my research and I think I'm a little too thick in the weeds with extraneous details to see it cleanly. If necessary, I can add some more conditions on ...
-3
votes
2answers
33 views

A Co-finite topology on $X$ is a discrete topology on $X$?(If $X$ is a finite set) [on hold]

So We need to show that our topology is power set of $X$. how can I proceed?
0
votes
1answer
32 views

$\mathcal{M}= \lbrace A \subseteq X | A \: \mbox{or} A^{c} \: \mbox{is numerable} \rbrace$ is a $\sigma$ algebra generated by a singleton family

Let $X \neq \emptyset$ and $$\mathcal{M}= \lbrace A \subseteq X | A \: \mbox{or} A^{c} \: \mbox{is numerable} \rbrace.$$ I want to prove that $\mathcal{M}$ is the $\sigma$-algebra generated by the ...
1
vote
1answer
39 views

what is the explicit definition of $K_\lambda$-filters?

I have a doubt with the definition of a concept: $K_\lambda$-filters. We say that a filter $\mathcal{F}\subseteq\mathcal{P}(\omega)$ is a $K_\lambda$- filter if it is $\textbf{generated}$ by less ...
3
votes
1answer
41 views

$C([0, 1],[0, 1])$ is dense in $\Pi_{[0, 1]}[0, 1]$?

Is the space $C([0, 1], [0, 1])$ of continuous functions $[0, 1] \to [0, 1]$ dense in the space $\Pi_{[0, 1]}[0, 1]$ of all such functions regarding pointwise convergence?
3
votes
1answer
23 views

Express the quotient of $\mathbb{R}^2/\mathbb{Z}^2$ by the relation $(x,y) \equiv (-x,-y)$ as the quotient of a polygon

This is another old qualifying exam question, and I'm kinda grasping at straws for this one. Let $\mathbb{T}^2$ denote the quotient space $\mathbb{R}^2/\mathbb{Z}^2$, and let $M$ denote the ...
0
votes
1answer
25 views

Totally disconnected but not totally separated

A topological space $X$ is totally disconnected if all of its components are singeltones, and $X$ is said to be totally separated if all of its quasi-components are singeltones. The set of rational ...
0
votes
1answer
28 views

Is a Co-finite Topology( Finite Complement topology) on a given set Unique for that set?

I was going through the definition given in J.R munkres: According to the definition, it seems to me that for every set $X$, there is a unique Co-Finite. Because it has been specified that collection ...
4
votes
0answers
69 views

Homotopy Lifting Problem

$X, \tilde{X}$ are hausdorff, path connected spaces $p:\tilde{X} \rightarrow X$ is a local homeomorphism Under these assumptions we have that: LEMMA If $f:Y \rightarrow X$ is continuous, and $Y$ ...
1
vote
1answer
29 views

Distance of a point from a set in a metric space .

$\mathbf {The \ Problem \ is}:$ Give an example of a metric space $X$, a point $x_0$, and a set $A \subset X$ so that $\operatorname{dist}(x_0,A) = 1$ where $\operatorname{dist}(x_0,A) =\operatorname{...
3
votes
2answers
53 views

Prove that $d$ is a metric on $X$.

Let $X$ be a set of real sequences and $d(x,y)=\sum_{k=1}^{\infty}\dfrac{1}{2^k}\cdot \dfrac{\vert a_k-b_k \vert}{1+\vert a_k-b_k\vert}$ with $x=(a_k)_{k \in \mathbb{N}}$, $y=(b_k)_{k \in \mathbb{N}} \...
0
votes
1answer
30 views

Topology Theorem on closure of a set [on hold]

If A is a subset of a metric space X ,then A closure is a closed set and is a subset of every closed set containing A
0
votes
1answer
37 views

The inclusion of a dense subset into a space is an epimorphism.

Let $A$ be a dense subset of a topological space $X$. Then the inclusion map $i:A\to X$ is an epimorphism. Is this true? If so, how do I prove it? I know the set up. Suppose $j,h:X\to Y$ are ...
-2
votes
1answer
36 views

Convergence of two arbitrary points in compact connected, metric space [on hold]

Let $X$ be a compact connected, metric space. Can we say, for any two-point $x,\,y\in X$ there is a sequence $\{a_i\}$ from $x$ to $y$ such that $a_0=x,\, a_i\in B_{d(x,\,y)}(y),\, \lim_{i\to \infty}...
0
votes
2answers
46 views

If $p$ is a regular covering then so is $q$.

If $q:X\to Y$, $r:Y \to Z$, and $p=r \circ q : X \to Z$ are all covering maps, with $Z$ locally path-connected, and if $p$ is a regular covering then so is $q$. Note 1. The condition $q$ is a ...
3
votes
2answers
36 views

Continuous Bijection of Top Spaces

It is well known that a continuous bijection of compact hausdorff topological spaces is a homoemorphism. I am wondering, is it true that a continuous bijection of compactly generated spaces is a ...
3
votes
2answers
40 views

How big can the Hausdorff dimension of the closure of a smooth curve be?

Consider curves in $\mathbb{R}^n$. Smooth curves have Hausdorff dimension $1$. The closure of a smooth curve can have Hausdorff dimension $> 1$. (For example, a curve dense in a torus.) How big ...
-1
votes
1answer
50 views

Definition of finite topology? [on hold]

Consider topology $T$ on $X$ where $X$ is a set given to us. When do we say topology is finite? When $X$ is finite? or when $T$ contains finite number of open sets? or both when $X$ is finite as ...
3
votes
2answers
68 views

Is the following set topology?

Consider a potential topology $T$ on $\mathbb R^3$ (three dimension space.) $T$ contains sets $$\{(x,y,z) \mid x^2 + y^2 + z^2 \leq r\},$$ where $r\in\mathbb R^{\geq 0}.$ Since we know that that ...
0
votes
1answer
70 views

A Direct Proof that a Topological Space with Ininitely Many Components is Disconnected

It's straightforward to prove that for a topological space, connected $\iff $ having one component. It then follows by contra-positive that disconnected $\iff$ having more than one component. In ...
0
votes
2answers
25 views

Is Sorgenfrey Line a discrete space?

The base of Sorgenfrey line consists of all sets of the form [a,b). By this all sets of the form [a,b) are clopen. How can we prove that the Sorgenfrey line is not discrete i.e. that the singelton {x} ...
2
votes
1answer
51 views

To prove topologically not same

The sets $A=\{ (x,y) \in \mathbb{R} \;|\; xy=0 \text{ and } x+y \geq 0 \}$ $B=\{ (x,y) \in \mathbb{R} \;|\; xy=0 \} $ are not homeomorphic. In $A$ if we remove the origin it becomes $2$ pieces, ...
0
votes
2answers
21 views

In general topological spaces (X,T) is it true that “S is with no isolated point = Every point of S is a limit point of S”?

In general topological spaces (X,T) (not only in metric spaces), let S be a non-empty subset of X, is it true that "S is with no isolated point = Every point of S is a limit point of S"? It seems ...
0
votes
0answers
32 views

Definitions for regular submanifold in terms of level sets?

From this question, we have: Let $S$ be a subset of a smooth $n$-manifold $M$. Then $S$ is an embedded $k$-submanifold of $M$ if and only if Every point $p\in S$ has a neighborhood $U\subset M$ ...
0
votes
0answers
33 views

Is the zero set of coordinate function a regular level set (regular zero set)?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
2
votes
1answer
26 views

A basic question about projection map with quotient space

I have a true-false statement as follows: Suppose $X$ is a topological space with a quotient space $Y=X/{\sim}$ and projection map $p:X\to Y$. If $U\subseteq X$ is open, then $p(U)$ is open. I think ...
2
votes
2answers
76 views

Verification of proof that if $S$ is connected then the closure of $S$ is connected.

I attempted to prove this theorem by contrapositive. Suppose $\overline{S}$ is disconnected. Then $\overline{S}=A\cup B$ such that $\overline{A}\cap B=A\cap \overline{B}=\varnothing$ and $A$ and $B$ ...
0
votes
2answers
75 views

How do we define convergence of a sequence in $\mathbb R\cup\{-\infty,+\infty\}$

Can we define convergence of a sequence in the set $\mathbb R\cup\{-\infty,+\infty\}$?How do we define that?Do we require metric space for this convergence?
1
vote
2answers
46 views

Show that a singleton $\{x\}$ is negligible

With the following definition of negligible set : $S \subseteq \mathbb{R}$ is negligible if $$\forall \varepsilon > 0 \hspace{0.2cm} \exists I_{k} : S \subseteq \bigcup\limits_{k \in \mathbb{N}}I_{...
1
vote
1answer
37 views

Prove that if $a,b\in R \Rightarrow \operatorname{frontier} (a,b)=\{a,b\}$

This would be the general case because I want to know how to prove that the frontier from(0,1)={0,1}. so, if i know the general case, i will know the particular case. Def. if A $\subset R \Rightarrow ...
3
votes
1answer
30 views

An injective neighborhood of a compact subset

I want to prove the following statement : Let $X$ be a (Hausdorff) space, $Y$ a metric space, $f : X \to Y$ a continuous function, $K$ a compact subset of $X$. If $f$ is injective on $K$, and if ...
-1
votes
2answers
59 views

Why does Munkres write that $f$ is bijective and continuous?

I have some doubts abot Munkres's topology My doubts: here $f([0,\frac{1}{4}))$ is not open in $S^1$ because $f([0,1/4))$ lying in the first quadrant. we know that continuous image of open set is ...
0
votes
2answers
39 views

Compactness Argument

Consider the inclusion chain of real valued intervals $\displaystyle \bigcup_{k=1}^\infty \left( a(k),b_\varepsilon(k)\right)\supset \bigcup_{k=1}^\infty (a(k),b(k)]\supset (a,b]\supset [a_\...
0
votes
1answer
29 views

Rational topological basis for Euclidean topology - topology without tears 2.2.3

Context: self-studying topology without tears, now at question 2.2.3. Question: Let $\mathcal{B}$ be the collection of all open intervals $(a,b) \in \mathbb{R}$ with $a \lt b$ and $a, b$ rational ...