0
votes
1answer
56 views

Properties of preimages and intersections of sets

I am working through Bert Mendelson's "Introduction to Topology" and am having some trouble with proofs. The text in well presented but to get a proper understanding I am working through the ...
1
vote
2answers
20 views

Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
3
votes
1answer
58 views

Sum of Neighborhoods of Zero

When do two neighborhoods of zero over a topological vector space add up as: $$aN+bN=(a+b)N\quad a,b\geq 0$$ I could imagine something like balanced might suffice... The problem is that I'd like to ...
1
vote
1answer
55 views

Regarding open subsets in topology

(Probably due to my lack of experience with the subject, I see that my question is horribly written. If you are to answer, a beginner-friendly explanation of the basis of a topology and the topology ...
5
votes
5answers
493 views

Why is that *any* union of open sets is open but only *finitely many* intersections of open sets is open?

I understand that when we talk about union of open sets, we introduce an index set which can be countable or uncountable. But could I not do the same for the intersection of open sets too?
1
vote
1answer
46 views

Noetherian toplogical space exercise

Let $X$ be a noetherian topological space. Prove the following statements: (a) If $F \subset X$ is closed, then there exist $n \in \mathbb N$ and irreducible closed subsets $F_1,\ldots,F_n \subset ...
5
votes
0answers
89 views

Show that it is an algebra.

This excercise is a little struggling for me. The part I need help with is showing that $D$ is closed under complements. Let $C$ denote the collection of all intervals on $\mathbb{R}$, including ...
1
vote
2answers
44 views

Construction of a small but fat set? [duplicate]

Is it possible to find a subset $A$ of the real line $\mathbb R$ such that the Lebesgue measure of $A$ minus its interior is positive ?
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
29 views

Cardinal number for a family of subsets to be a topology

Problem Let $\kappa$ be a cardinal number and let $$\tau_{\kappa}=\{U \in \mathcal P(X) : X \setminus U \space \text{has cardinal at most} \space \kappa\} \cup \{\emptyset\}$$ Determine the ...
0
votes
1answer
38 views

Find the topologies of sets of at most four elements

Problem Find all the topologies of the sets with at most four elements. The attempt at a solution I've tried to divide the problem into five cases according to the cardinal of the set I know that ...
0
votes
3answers
49 views

Closure of a subset of a metric space is closed

From definition, if $X$ is a metric space, if $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\overline{E}=E \cup E'$. I need to ...
0
votes
0answers
33 views

What does “single set” mean in this context?

I encountered this problem in Munkres topology. Let $X_1 , X_2$ denote a single set in topologies $\tau_1$ and $\tau_2$, respectively; let $Y_1 , Y_2$ denote a single set in the topologies $U_1, ...
0
votes
1answer
99 views

Question about proof on basis

I found this proof online, but I have a bit of trouble understanding it. Question: Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an ...
3
votes
2answers
46 views

The union of all the open sets in a family of topologies

I'm starting studying topology for the first time and my teacher just wrote this. I just don't understand the last line: Let $\{\tau_\alpha\}$ be a family of topologies on X. [...] To say that ...
1
vote
0answers
20 views

Non-section representation of an intersection of sets

Let $X,\bar X,Y$ be arbitrary sets and $A\subseteq X\times Y$, $\bar A\subseteq \bar X\times Y$ be arbitrary as well. Denote: $$ A_x :=\{y\in Y:(x,y)\in A\} $$ and similarly for $\bar A$. Consider a ...
2
votes
1answer
34 views

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets

How can I prove that $\mathbb R$ contains no more then $\mathfrak c$ $F_\sigma$ sets? (or equivalently, that $\mathbb R$ contains no more then $\mathfrak c$ $G_\sigma$ sets? The more general ...
0
votes
2answers
242 views

Why it is always circle to represent a Set?

When we draw a Venn diagram, we use circle to represent a Set. We can use any closed plane figure but most of the time it is a circle. Why? are there any specialty about that?
1
vote
1answer
38 views

Calculate the number $o(\mathbb{R})$ of open subsets of the real line. [duplicate]

Calculate the number $o(\mathbb{R})$ of open subsets of the real line. I know that the answer is $\mathfrak{c}$ but I don't know how my lecturer got this. I am doing an introductory topology course, ...
3
votes
1answer
23 views

Language clarification in an article about filters

I started reading these notes. After enumerating four properties of a filter $\mathcal F$ in a topological space $(X,\tau)$ (1) $X\in\mathcal F$; (2) $V\in\mathcal F\wedge V\subseteq ...
2
votes
3answers
60 views

Metric Space, Induced topology

I'm trying to show that the metric topology is indeed a topology. To do so, I want to show the following three statements are true: $\emptyset$ and X are in $\tau$ finite intersections of open sets ...
0
votes
1answer
59 views

Set theoretic disjoint union

In this page we read Let $X$ and $Y$ be topological spaces and $Z := X \coprod Y$ be a set-theoretic disjoint union. We wish to define a topology on $Z$ in a most natural way. Definition. ...
0
votes
1answer
15 views

Continuity of restriction map

Consider a map $f:X\rightarrow Y$ and suppose $X=\bigcup_i U_i$ is a union of open subsets. Prove that if all the restrictions $f_i=f|_{U_i}:U_i\rightarrow Y$ are continuous, then $f$ is continuous. ...
2
votes
1answer
28 views

Continuity and Leximin

Consider a relation $\geq$ over the set of real-valued vectors. We say that $\geq$ is continuous if for any positive integer $n$, and any $\pi\in\mathbb Z^n,u\in\mathbb R ^n$ we have that the sets ...
0
votes
0answers
23 views

Are all subbasis subsets of basis?

In topology, each element of basis $\{B_k\}$ can be expressed as finite intersections of elements of subbasis, i.e. $B_k=S_{n_1}\cap ...\cap S_{n_m}$ Does the meaning of "finite intersections" also ...
3
votes
1answer
42 views

Is the collection of atlases on a set $X$ a set?

Well, the title says it all. I need to know if i can view the collection of all atlases on a given set $X$ as a ordinary set. Is this possible ? All the atlases are only topological atlases, no ...
0
votes
3answers
30 views

If $A_1\cap…\cap A_n \neq \emptyset$, does $(A_1\cap…\cap A_n)^{c} =A_1^{c} \cup … \cup A_n^{c} = \emptyset$?

If I have some collection of sets such that $A_1\cap...\cap A_n \neq \emptyset$, then what happens if I apply the complement (denoted by superscript c) to both sides? i.e., $(A_1\cap...\cap A_n)^{c} ...
0
votes
1answer
49 views

Basic topology questions with cantor's set

I have 3 questions in toplogy, one of which I managed to solve (but would appreciate input regardless) and 2 which are more difficult. I'd like a push in the right direction. Define $K$ as ternary ...
-1
votes
1answer
62 views

Hausdorff topologies on the natural number set are sigma algebra

Is it true that if I add the Hausdorffness condition to any topology on $\mathbb{N}$, then it is a $\sigma$- algebra on $\mathbb{N}$? Once I have tried to prove this, I think that compactness is also ...
5
votes
2answers
83 views

Denseness of a set, whose complement is known to be dense.

Do you think in general that if say $U\subset X$ was dense in $X$, then if we let $V=X−U$, but we know $V$ is of higher cardinality than $U$, does that imply that $V$ must be dense?
0
votes
4answers
94 views

How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
0
votes
4answers
69 views

Are the axioms of a topological space superfluous?

A topology on a set $X$ is a family $\mathcal{T}$ of subsets of $X$, which are open sets and satisfy: (1) $\emptyset, X \in \mathcal{T}$. (2) Any union of elements of $\mathcal{T}$ belongs to ...
3
votes
2answers
55 views

Proof that a given projection map restricted to a subset is closed.

$\pi_{1}:\mathbb{R}^2\rightarrow\mathbb{R}, (x,y)\mapsto x$ is a projection map from $\mathbb{R}^2$ with the standard eulcidean topology, $\mathscr{T}_E$ to $\mathbb{R}$ with it's usual euclidean ...
0
votes
1answer
31 views

Opnennes of two similar sets

So we have the following sets, and the question is whether they are open (to be more specific, this is asked to show using concepts of relative topology). Note that $\overline{B} = \{b=(b1,b2) \in ...
2
votes
3answers
241 views

Inverse limit of an inverse system of topological spaces

Given an inverse system $\mathcal G=\{X_i\}$ of topological spaces over some directed set $I$. If $X=\prod\limits_{i\in I}X_i$, the inverse limit $X^*=\varprojlim X_i$ of $\mathcal G$ is a subspace ...
2
votes
1answer
41 views

Cartesian Product of Linearly ordered space and an example

The base of the cartesian product of linearly ordered spaces $A$ and $B$ is the form $\{U\times V : U\text{ open in }A, B\text{ open in }B\}$. By using this, we condiser the Long Line Topology which ...
1
vote
2answers
41 views

Point as an element of an affine space vs point as an element of a topological space?

I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong ...
2
votes
3answers
52 views

Why is the infinite union of cofinite sets necessarily cofinite with regard to the cofinite topology of the reals?

The open sets of the cofinite topology on $\mathbb{R}$ are defined as all sets in $\mathbb{R}$ whose complement is finite. Can someone point out the error in my logic with regards to the union of any ...
1
vote
2answers
186 views

A Theorem About Compactness and

My first exposure to any sort of topology is from Spivak's Calculus on Manifolds. I think I understand compactness conceptually, I'm just finding the rigor a little bit elusive. My first question ...
2
votes
2answers
169 views

Why is this not a closed set?

I can clearly see that I can create disks around the entire perimeter of this triangle where there's at least one point in the disk that's in the triangle, and outside of the triangle. So why is ...
0
votes
2answers
53 views

Definition of $\omega_1$, comparing it to $2^\mathbb{N}$?

I'm taking an Intro to Topology class, and we just started defining ordinals. We defined finite ordinals as: \begin{align*} 0 & = \varnothing \\ 1 & = \{0\} \\ 2 & = \{0,1\} \\ & ...
10
votes
2answers
141 views

A Question regarding disjoint dense sets

If we take the standard topology on $\mathbb{R}$ we can easily find two disjoint sets that are dense, namely $\mathbb{R}\setminus\mathbb{Q}$ and $\mathbb{Q}$. Similarily, if we take the same topology ...
1
vote
1answer
30 views

Continuity: Topology by Munkres

I can't seem to convince my self of this equality, assume that $f|U_{\alpha}$ is continuous for each $\alpha$, then if V is open in Y for Y being a topological space then, $f^{-1}(V) \cap ...
2
votes
3answers
207 views

Proving that cardinality of the reals = cardinality of $[0,1]$

Homework problem, intro to topology. Here's what I've done so far. Am I on the right track? And, how would you advise me to proceed from here? I have already established that $\left |[0,1] \right | = ...
0
votes
1answer
41 views

Problem involving an infinite lattice grid

I'm stuck on this problem for Intro to Point-Set Topology.... I'm given that a submarine starts somewhere in $\mathbb{R}^2$ and moves in a straight line at constant velocity, in such a way that at ...
2
votes
3answers
55 views

does the domain can be considered as subset of it image under 1 to 1 function?

Let $f\colon X \to X$ be a one-to-one function and let $A \subseteq X$. Does $A \subseteq f(A)$? I ask because I found a step which not clear to me in this paper ...
5
votes
4answers
436 views

Do these theorems about power sets hold for the empty set?

From the definition of the power set as the set of all subsets of a given set, I realize that $\mathcal P(\varnothing) = \{ \varnothing \}$, in other words, the power set of the empty set is the set ...
2
votes
2answers
68 views

The difference between a finite set and an ordered $n-$tuple? Proving the set of all finite subsets of a countable set is countable.

For my point-set topology class, I'm working on proving the theorem: The set of all finite subsets of a countable set is countable. Please don't post the proof of the theorem. The proof was easy for ...
0
votes
1answer
48 views

Intuition behind Topological Spaces, Intersection Property

I am in an introduction to Topology course and I was unsure about the intuition I have on a couple questions required to be handed in. So here goes: Let X be R, the reals, and let Omega consists of ...
8
votes
2answers
88 views

How should one think about results that depend on AC?

I just encountered this: "(Theorem of A. H. Stone) Every metric space is paracompact... Existing proofs of this require the axiom of choice... It has been shown that neither ZF theory nor ZF ...