Tagged Questions
4
votes
2answers
82 views
Valid Proof that the Irrationals are Uncountable?
So I originally wanted to prove that the reals are uncountable, but the best solution I came up with was to prove the irrationals are uncountable so therefore the reals must be as well. I suppose my ...
7
votes
2answers
62 views
Closed curves in 3D
Is $R^3$ a union of disjoint closed curves? (Obviously $R^3$ minus a line is). Is this a classical problem?
2
votes
1answer
33 views
When in topology is $A = f^{-1} \circ f[A]$ or $B = f \circ f^{-1}[B]$ true, for an $f$ which is not one-to-one?
I'm having a bit of trouble with an example problem in the topology book I'm reading. It's problem #11 (pp 104) of the "Solved Problems" section of Chapter 7, of the Schaum's Outline for "General ...
1
vote
1answer
26 views
On the limit of a Minkowski sum
Consider an open set $\mathcal{O} \subseteq \mathbb{R}^n$.
I am wondering if the set $$ \mathcal{S} := \lim_{k \rightarrow \infty} \ \mathcal{O} + \frac{1}{k} \mathbb{B} $$
is open or closed.
With ...
9
votes
5answers
263 views
What is the canonical definition of an open set?
The definition of an open set that I see in most topology texts(like the ones found in Topology by Munkres and another w/ the same title by Hocking & Young, or Basic Topology by Armstrong) is that ...
0
votes
1answer
81 views
Are there only 2 clopen sets on real plane?
How can I prove that the only open and closed sets on the real plane are empty set and real plane itself? Preferably by using order theory.
Thanks.
22
votes
1answer
532 views
Showing a filter on the Power set of $\mathbb{Z}$ is a one point Filter
Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...
4
votes
2answers
52 views
Intuition behind the difference between derived sets and closed sets?
I missed the lecture from my Analysis class where my professor talked about derived sets. Furthermore, nothing about derived sets is in my textbook. Upon looking in many topology textbooks, few even ...
2
votes
1answer
71 views
How to prove that every space can be decomposed into scattered and perfect subspace
How to prove the following claim:
A space $X$ can be represented as the union of two disjoint subset $A$ and $B$, where $A$ is a scattered and $B$ satisfies $B=B'=:\{x: x \text{ is the accumulation ...
6
votes
1answer
38 views
A question on derived set
Let $A$ be a set and $A'=\{x: x \text{ is the accumulation point of A}\}$, which is called the derived set of $A$. So, we can definine $A''$ of $A'$.
Then do we have $A''\subset A'$?
Thanks for ...
4
votes
2answers
82 views
are “all nets in $X$” well defined?
Denote $f:X\to Y$ as a function between topological spaces $X$ and $Y$. One good way for determining whether $f$ is continuous is to check the following statement.
$f$ is continuous iff for every ...
3
votes
1answer
35 views
$\tau:=\{Y\in P(X) | A\subseteq Y\}\cup\{\emptyset\}$ topological space [duplicate]
Let $X$ be a set and $A\subseteq X$
I would like to show that $\tau:=\{Y\in P(X) | A\subseteq Y\}\cup\{\emptyset\}$is a topological space on $X$ and afterwards I would like to describe the closure ...
7
votes
2answers
67 views
Need help on Furstenberg's proof on the infinitude of primes
I have a question on this proof given by Furstenberg proof on the infinitude primes. I am a non-mathematician with some basic knowledge on set theory and topology.
Define for $a,b\in\mathbb{Z}$ where ...
6
votes
5answers
138 views
What is the difference between $f(f^{-1}(U))$ and $f^{-1}(f(U))$?
What is the difference between $f(f^{-1}(U))$ and $f^{-1}(f(U))$.
Now i Think if $f$ is onto then $f(f^{-1}(U)) = U $ while $f^{-1}(f(U))=U$ no matter what.
why don't I have a clear picture in my ...
8
votes
2answers
262 views
How many paths exist between two points in the plane?
Fix distinct $a,b \in \mathbb{R}^2$. In terms of cardinality (say, beth numbers), how many distinct continuous functions $f : [0,1] \rightarrow \mathbb{R}^2$ satisfying $f(0)=a, f(1)=b$ are there? ...
3
votes
3answers
79 views
What sets contain $\infty$ and $-\infty$ and why are the Integers closed?
So I'm currently studying from Rudin's Principles of mathematical analysis or colloquially "Baby Rudin" and have stumbled into the second chapter namely basic topology. He lists some sets and states ...
4
votes
1answer
35 views
Cardinality of cofinite topology
Does it make sense to ask about cardinality of closed finite topology defined on the set of Natural Numbers? Is it countable or uncountable? Is it possible to prove that it is countable?
Further, if ...
4
votes
3answers
64 views
Are these open sets?
This is the question:
Which of these sets are open sets on the lower limit topology on $\mathbb{R}$, whose basis elements are $[a,b),a<b$?
$$[4,5)\qquad\left\lbrace3\right\rbrace\qquad ...
1
vote
1answer
49 views
How to read a chapter about connectedness for topological spaces as if you only want to know things about metric spaces?
I am reading chapter 12 of the book "Introduction to Metric&Topological Spaces" written by Wilson A. Sutherland, because I follow a course is about metric spaces. I missed some classes.
The ...
2
votes
3answers
150 views
3
votes
1answer
46 views
Do we have $\overline{\bigcup\{A_\beta: \beta<\alpha\}}=\bigcup\{ \overline{A_\beta}: \beta<\alpha\}$
Let $\alpha$ be any ordinal, and for the family $\{A_\beta: \beta< \alpha\}$, we have $A_{\beta_1}\subset A_{\beta_2}$ when $\beta_1<\beta_2<\alpha$, then do we have ...
0
votes
2answers
81 views
Find all properties of this set $S$ in the metric space $(\mathbb{R},\rho)$
Set $S=(0,1]\cup([-\sqrt{2},\sqrt{3})\cap\mathbb{Q})$ in metric space $(\mathbb{R},\rho)$ with standard metric $\rho(x,y)=|x-y|$.
Find the following sets:
$S'$, $\text{Closure}(S)$, ...
1
vote
1answer
86 views
Is this space countably compact
Let $X$ be a Tychonoff countably compact space and $A$ is a subapce of $X$ such that for any countable $B \subset A$ we have $\overline{B} \subset A$. My question is this:
Is this subspace $A$ ...
2
votes
1answer
103 views
Bounded metric space $(X,\rho)$ question for any $S\subset{X}$?
Prove that in any metric space $(X,\rho)$ for any $S\subset{X}$, we have $bd(bd(S)) = bd(bd(bd(S)))$, while not necessarily $bd(S)=bd(bd(S))$
Prof's Hint (first show that a boundary of a closed set ...
1
vote
1answer
60 views
Relation between convergence class and convergence space
A convergence class is defined from nlab
A convergence space is a set $S$ together with a relation $→$ from $ℱS$ to $S$, where $ℱS$ is the set of filters on $S$; if $F→x$, we say that $F$ ...
3
votes
2answers
89 views
Why is a rectangle not a neighborhood of its corners?
I'm trying to puzzle out a statement given in the Wikipedia article on topological neighborhoods, which uses this definition:
If $X$ is a topological space and $p$ is a point in $X$, a ...
3
votes
2answers
71 views
Second-countable implies separable/Axiom countable choice
Let $(X,\mathscr T)$ be a topological space, and $(B_n)_{n\ge1}$ a countable basis for X. Under this assumptions, X is separable.
The proof of this assertion is as follows:
We can assume without ...
2
votes
4answers
114 views
Is there a good visual aid or picture to help understand openness and closedness?
I'm struggling to grasp the idea of open, closed, clopen and not open and not closed sets in a more formal approach like how it's described in a math analysis class. Is there a good picture somewhere ...
1
vote
2answers
54 views
How to understand closed subsets of limit ordinal?
On page 20, Constructibility, K.J.Devlin,
(Let $\alpha$ is a limit ordinal)A set $A \subseteq \alpha$ is closed, iff $\bigcup A \cap \gamma \in A$ for all $\gamma < \alpha$. Equivalently, if we ...
0
votes
1answer
75 views
Showing the convergence of a sequence of compact nonempty sets
Given convergent sequences of compact sets $\{A_k\}$ and $\{B_k\}$ with $\lim_{k \rightarrow \infty} A_k = A_{\infty}$ and $\lim_{k \rightarrow \infty} B_k = B_{\infty}$, $A_k \cap B_k \neq \emptyset ...
3
votes
1answer
74 views
Minimal Connected Set containing a Closed Connected Set in a Compact Space
This question came from Dugundji's $\textit{Topology}$: Given a compact, connected space $X$, let $A \subset X$ be closed. Prove that there exists a closed, connected set $B \subset X$ such that $A ...
2
votes
1answer
52 views
Precedence of $\times$ and $\cup$.
In topology, there is a very strong need of describing subsets of some product $X\times Y$ by means of unions and products.
For example, this is a very convenient way of describing subsets of the ...
1
vote
1answer
88 views
Foundational problem with set theory notation, and writing proofs out in the language of set theory
Let $E \subseteq \mathcal{P}(X)$ be a family of sets, then there exists a smallest toplogy containing $E$, i.e.
$$
O_E := \bigcap_{E \subseteq O, O \textrm{ is topology}} O.
$$
And now I read about a ...
3
votes
1answer
76 views
A question on linear ordered space
A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$. And we know every space contains a dense left-separated subspace.
My question ...
6
votes
1answer
146 views
Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems
There are some set systems with algebraic titles, such as "field",
"algebra", "ring" and "semi-ring" (and possibly other titles), in
their names. Examples are
a sigma field (aka sigma algebra, ...
4
votes
1answer
159 views
When does it make sense to define a base of a set system?
In a topology, a base is defined to be a class of subsets such that
every open set is the union of some members of it.
In a convexity
structure, a base is defined to be a class of subsets ...
2
votes
2answers
85 views
When does it make sense to define a generator of a set system?
In a set system, such as a topology, sigma algebra or
convexity structure, a generator is defined to be a class of
subsets such that the given set system is the coarsest such set
system ...
2
votes
3answers
166 views
Cardinality of all dense and countable sets of $\mathbb{R}$
What is the cardinality of the following set:
$$\mathbb{A}:=\{A \ : A\subseteq\mathbb{R} \ \ \text{dense and countable}\}$$
(Is $\mathbb{A}$ a separable space?)
Thank You!
2
votes
1answer
111 views
What is a saturation of a set?
So, I encountered in my topology book the saturation of a set and In my first language the translated word is rarely used and those papers I found who use it don't explain it since it seems to be ...
1
vote
3answers
165 views
What is the interior of a singleton?
I was wondering what can be said about the interior of $\{{4}\}$, the empty set?
The interior of a set $A$ is the largest open set contained by $A$. Hence, if the set at hand is a singleton, then ...
1
vote
2answers
63 views
Can anyone clarify how a diverging sequence can have cluster points?
$p$ is a cluster point of $S\subset M$ if each neighborhood of $p$ contains infinitely many points. Here is my confusion, a cluster point is also a limit point of $S$, right?
If so, then how does the ...
0
votes
1answer
218 views
Given, the cartesian product of two non-empty sets A and B (subsets of a metric space M) is sequentially compact, show that A and B are compact
We know that if $A$ and $B$ are compact (assuming A and B are non-empty), then the Cartesian product $A \text{x} B$ is compact. But how do you go the other way round.
We have to show that any ...
11
votes
3answers
316 views
Cantor set and countability.
The Cantor set is closed, so its complement is open. So the complement can be written as a countable union of disjoint open intervals. Why can we not just enumerate all endpoints of the countably ...
1
vote
1answer
37 views
“Agreement Domain” of Function Families
Given a family of functions $\{f_i : X \to Y\}_{i \in I}$ between topological spaces $X$ and $Y$, I define an operation $\bigcap\limits^{\scriptscriptstyle\text{dom}}$ on the family such that ...
3
votes
3answers
79 views
What is the negation of this statement?
Let $(K_n)$ be a sequence of sets.
What is the negation of the following statement?
For all $U$ open containing $x$, $U \cap K_n \neq \emptyset$ for all but finitely many $n$.
1
vote
4answers
734 views
Prove that the intersection of a finite number of open sets is open.
More specifically, let $O_1, . . . , O_n$ be a finite collection of open subsets of the continuum, $C$. Then the intersection $O_1 ∩ · · · ∩ O_n$ is open as well. I think it is possible to do it ...
0
votes
2answers
89 views
Alternative definition of density
Let $E$ be called dense in $\mathbb{R}$ if and only if $\text{int}(\mathbb{R} \setminus E)=\emptyset.$
Let $x \in\text{int}(\mathbb{R} \setminus E)=\emptyset$. Then for $\epsilon >0$, ...
1
vote
1answer
64 views
Conditions to ensure this union is closed
Let $(M,d)$ be a metric space and let $\{S_n\}_n$ be a countable collection of non-empty closed and bounded subsets of $M$
Are there any additional conditions on the collection$\{S_n\}_n$ to ensure ...
3
votes
5answers
190 views
Is a countable set $A$ dense in an uncountable set $B$ by analogy of $\mathbb{Q}$ being dense in $\mathbb{R}$?
Let $A$ be any countable set and $B$ any uncountable set with the same infinity as $\mathbb{R}$. Then we must have a one-to-one map $\phi:A\rightarrow\mathbb{Q}$. So, can we say by all this that $A$ ...
1
vote
5answers
204 views
Is there a “natural” topology on powersets?
Let's say a topology T on a set X is natural if the definition of T refers to properties of (or relationships on) the members of X, and artificial otherwise. For example, the order topology on the set ...
