1
vote
2answers
55 views

What's the diference between $A<\infty$ and $A<\aleph_0$?

In my topology class the teacher gave some examples of topologies, and I'm trying to prove that they really are topologies. If $X$ is a set then: $\mathcal C=\{A:\# (X-A)<\infty\}$ is a topology ...
0
votes
0answers
49 views

Is this sentence OK?

I'm starting to write a paper. This is the sentence which I want to put first in the paper. It is well known that diagonal properties are useful in estimating certain cardinal invariants of a ...
2
votes
1answer
59 views

cardinality of the set of all dense subsets of $\Bbb R$

Let $$A=\{X \subseteq \mathbb R : \operatorname{cl}(X)=\mathbb R\}$$ Prove that the set $A$ and $P(\mathbb R)$ have the same cardinality. Well, the first thing it came to my mind was the injective ...
2
votes
1answer
55 views

A question on a closed and discrete subset

A generalization of Lindelöf number is extent, defined as follows: $e(X)=\sup \{|D|: D \subset X, D \text{ is closed and discrete } \} + \omega$. The weight of $X$ is defined by $w(X)= \min\{|\mathcal ...
3
votes
2answers
175 views

What is the cardinality of $[a,b] $?

It is a well-known fact that any open interval $(a,b)$ has the same cardinality as $\mathbb R$; that is, there exists a bijection $f\colon(a,b)\mapsto \mathbb R$. What about the closed interval ...
1
vote
1answer
72 views

What is the cardinality of $X$?

Let $X=\{x\in D^\mathfrak{c}: 0<|\{\xi<\mathfrak{c}:x(\xi)=1\}|\le\omega_1\}$, where $D=\{0,1\}$. What is the cardinality of $X$? I think it is $\mathfrak c$, however I'm not sure. Also I don't ...
2
votes
0answers
52 views

A question on semi-stratifiable spaces

A space $(X,\tau)$ is called semi-stratifiable if there exists a function $g:\omega\times X\to\tau$ such that: (i) for any point $x$ of $X$ holds $\{x\}=\bigcap_{n\in\omega} g(n,x)$; (ii) for any ...
4
votes
0answers
76 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
0
votes
1answer
77 views

A conjecture on closed discrete subset

I am struggling with this old problem: Let $X$ satisfy countable chain condition(abbreviated as CCC) and $X$ has a regular $G_\delta$-diagonal. Then the cardinality of $X$ is at most $\mathfrak ...
1
vote
2answers
84 views

Cardinality of the class of $G_\delta$ subset of $\mathbb{R}$ of Lebesgue measure zero

Let $\mathcal{N}$ be the class of all subsets of $\mathbb{R}$ of Lebesgue measure zero and let $\mathcal{G}_\delta$ be the class of all $G_\delta$ subsets of $\mathbb{R}$. How do I show that ...
4
votes
1answer
74 views

If $X$ is Hausdorff and $|X|> \mathfrak{c}$, does $X$ always have a uncountable discrete subspace?

Let $X$ be a Hausdorff topological space with $|X|> \mathfrak c$. Does $X$ always have a uncountable discrete subspace? Thanks for your help.
8
votes
1answer
431 views

Tightness and countable intersection of neighborhoods

The following is a problem a colleague has encountered. He would like to know whether the following conjecture is right, wrong, or neither: Let $X$ be a topological space of countable tightness ...
5
votes
1answer
85 views

$\kappa\psi (x,X)\leq \psi (x,X)$

The $\kappa$-pseudocharacter $\kappa\psi (x,X)$ of a space $X$ at a point $x\in X$ is the smallest infinite cardinal number $\tau$ such that there exists a family $\gamma$ of $\kappa$-sets in $X$ ...
5
votes
2answers
80 views

Product of a family of spaces of countable tightness

I recently learned the concept of cardinal functions and some of the definitions and theorems are not clear to me. How can we prove this theorem? Finite family of compact spaces of countable ...
3
votes
1answer
59 views

“big” Hausdorff space with dense subspace of given cardinality

In a topology course we proved the following proposition: Let $A$ be an infinite set. Then there exists a Hausdorff space $X$ of cardinality $|\mathfrak{P}(\mathfrak{P}(A))|$ which contains a ...
3
votes
1answer
70 views

How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
1
vote
1answer
38 views

CCC and point separating weight

CCC means countable chain condition; A cover $\cal A$ of a set $E$ is separating if for each $p\in E$, $\bigcap \{A: A \in \mathcal{A}, p\in A\}=\{p\}.$ The point separating weight of $X$, denoted ...
3
votes
1answer
50 views

Inequality in cardinal function

How to prove that $|X|\le 2^{s(X)\psi(X)}$ by using the Erdős-Rado theorem when $s(X)=\psi(X)=\omega$? $s(X)=\sup \{ |D|: D \subset X, D \text{ is discrete} \} + \omega $ $\psi(X)= \sup\{\psi(p,X): ...
2
votes
1answer
51 views

A question on a set theoretic theorem

Erdős-Rado theorem: Let $\kappa$ be an infinite cardinal. Let $E$ be a set with $|E|>2^\kappa$, and suppose $[E]^2=\bigcup_{\alpha<\kappa}P_\alpha$. Then there exists $\alpha<\kappa$ and a ...
2
votes
1answer
176 views

Cardinality of the set of clopen subsets of a topological space

Is there some way to find the cardinality of set of all clopen subsets of a topological space, say, Cantor space, Baire space?
14
votes
1answer
299 views

Cardinality of a locally compact Hausdorff space without isolated points

I am interested in the following result: Theorem. A locally compact Hausdorff topological space $X$ without isolated points has at least cardinality $\mathfrak{c}$. To prove it, one can find two ...
3
votes
1answer
103 views

Cardinalities of topologies in which not each open set is a union of regular open sets

Suppose, a topological space $(X, \mathscr{T})$ consists of a set $X$ with the cardinality $\kappa$, and a topology $\mathscr{T}$ in which it is not true that each open subset of $X$ can be written as ...
3
votes
2answers
179 views

Cardinality of an infinite separable connected metric space is $2^{\aleph_0}$.

How to prove: Cardinality of an infinite separable connected metric space is $2^{\aleph_0}$. Thanks in advance!!
5
votes
2answers
441 views

In set theory, what does the symbol $\mathfrak d$ mean?

What's meaning of this symbol in set theory as following, which seems like $b$? I know the symbol such as $\omega$, $\omega_1$, and so on, however, what does it denote in the lemma? Thanks ...
5
votes
1answer
338 views

Is $2^\infty$ uncountable and is cardinality a continuous function?

I apologize if the title seems too vague, but this is how I was asked the question. So one of my friends intended to write an infinite sum like $\displaystyle \sum_{i=1}^{\infty} a_{2^i}$ . However, ...
2
votes
2answers
276 views

About Cardinality in Hausdorff Spaces

I have two problems: 1.- Let $X$ be a compact Hausdorff space, then $X$ has a basis with cardinality less than or equal to $|X|$. 2.- Let $X$ be a Hausdorff space and $D$ a dense subset in $X$, ...