0
votes
1answer
74 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
41 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
19 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
33 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
236 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
35 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
55 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
56 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
22 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
46 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
60 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
82 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
88 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
67 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
52 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
219 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
40 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
40 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
48 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
183 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
166 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
134 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
201 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
431 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
62 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
46 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
86 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 ...
1
vote
2answers
100 views

Difference between $R^\infty$ and $R^\omega$

I know $R^\omega$ is the set of functions from $\omega$ to $R$. I would think $R^\infty$ as the limit of $R^n$, but isn't that $R^\omega$? The seem to be used differently, but I can't tell exactly ...
1
vote
1answer
46 views

Baire Category Theorem: What should we really prove there?

I am reading about the Baire Category Theorem in Jech's book on set theory. 4.8: Baire Category Theorem: Let $D_0,D_1,\dots,D_n,\dots$, $n \in \mathbb{N}$, be open dense subset of $\mathbb{R}$. Then ...
2
votes
1answer
54 views

How to rigorously prove that these two sets have different order types?

Let $A$ and $B$ be two given ordered sets with the linear (or total) order relations $<_A$ and $<_B$, respectively. Then $(A,<_A)$ and $(B,<_B)$ are said to be of the same type if there ...
0
votes
1answer
35 views

Coincident boundaries of sets

At the end of my lecture, which introduced closed sets and boundaries of sets, my professor asked the following question: Let $S,T,V \subset \mathbb{R}^2$ such that $int(S) \not = \emptyset$, $int(T) ...
4
votes
0answers
47 views

Metric-like families of relations

Let $X$ be an arbitrary set and to start with, let us consider a relation $\leq$ on $X$ (that is $\leq$ is a subset of $X^2$) which is reflexive and transitive. such a relation is called a preorder. ...
1
vote
1answer
44 views

If a topological space $X$ has a countable basis. Then if we have an open cover of $X$, can this cover be refined to a countable one?

If a topological space $X$ has a countable basis. Then if we have an open cover of $X$, can this cover be refined to a countable one?
1
vote
0answers
17 views

Count the number of topological sorts for each poset [duplicate]

Count the number of topological sorts for each partially ordered set $(A,|)$, where (a) $A = (3, 5, 7, 11, 13, 16, 17)$ (b) $A = (1, 3, 9, 27, 81, 243)$ That is, you have to find the number of ways ...
0
votes
1answer
50 views

Verification - Are the intersections/unions of these sets convex/compact/bounded?

I'm looking to see if I have found the correct answers to these questions. Define for $a, b \in \mathbb{R^{2}}, d>0, r>0 $ the sets $V_{a;d} = \{x \in \mathbb{R^{2}}:\max\{|x_{1}-a_{1}|, ...
2
votes
2answers
85 views

Closure boundary interior sets

If we denote for a set $A$: $A^{o}$ the interior points set; $\overline A$ the closure and $\delta A$ the boundary set and $A'$ the set of cluster points , do the following hold (give counter-examples ...
1
vote
2answers
57 views

Is the intersection of a sequence of nested subspaces nonempty?

Say $X$ is a topological space (compact). if $\{ A_n \} $ is a collection of nonempty closed subsets of $X$ such that $A_{n+1} \subseteq A_n $ for all $n$, then does it follow that $ \bigcap_n A_n $ ...
0
votes
2answers
55 views

Why is the boolean closure of $F_{\sigma}$-sets not in $F_{\sigma}\cap G_{\delta}$?

In the Borel hierarchy, why is the boolean closure of $F_{\sigma}$ or $G_{\delta}$ equal to $F_{\sigma \delta} \cap G_{\delta \sigma}$? If I take the complement of an element in $F_{\sigma}$ I got an ...
0
votes
0answers
18 views

Hemicontinuity of multifunctions/correspondences that can map to the empty set

The Wikipedia article on hemicontinuity of multifunctions or correspondences does not make it clear whether the multifunction or correspondence $f : A \to 2^B$ (power set) is allowed to map to the ...
1
vote
1answer
91 views

A question about convex set

I need to prove the closed set $C\subseteq \mathbb{R}_{+}$ is a convex. And let $x$, $y$ be arbitrary given in $C$, I have proved that $1/2(x+y)\in C$. Then does this means $C$ is convex ?