2
votes
1answer
61 views

Cardinal characteristics

Assuming Continuum hypothesis is not true, How many cardinals $k$ exist which are $\aleph_1 < k < \mathfrak c$? Can I assume that there is a finite number of these cardinals or is there an ...
4
votes
1answer
88 views

How many subspace topologies of $\mathbb{R}$?

Say two subsets of $\mathbb{R}$ are equivalent if they are homeomorphic, with the subspace topology. How many equivalence classes are there? It's immediate that there are at least $\beth_0$ (we can ...
4
votes
1answer
58 views

Does the converse of Tychonoff's theorem hinge on the axiom of choice?

Tychonoff's theorem:$\phantom{---}$ If $A$ is a non-empty index set and $X_{\alpha}$ is a non-empty compact topological space for every $\alpha\in A$, then $X\equiv\times_{\alpha\in A} X_{\alpha}$ is ...
1
vote
1answer
64 views

Product of Borel $\sigma$-algebras vs Borel $\sigma$-algebra of product

If $X$ and $Y$ are topological spaces with associated Borel $\sigma$-algebras $\mathcal{B}_X$ and $\mathcal{B}_Y$, then the product $\sigma$-algebra $\mathcal{B}_X\otimes \mathcal{B}_Y\subset ...
2
votes
1answer
63 views

Suppose $X$ is a Hausdorff Lindelöf scattered space. Is $\xi(X)$ a successor ordinal?

Let $K$ be a (Hausdorff) scattered topological space and for each ordinal $\alpha$ denote by $K^{(\alpha)}$ the $\alpha$th derivative of $K$ by the Cantor-Bendixson derivation (i.e., define ...
1
vote
1answer
49 views

Closed subsets of $\beta \mathbb R$

Definitions. Suppose $X$ is a topological space. $w(X)=\min\{|\mathcal B|:\mathcal B$ is a base for $X\}+\omega$ $e(X)=\sup\{|D|:D\subseteq X$ is closed and discrete$\}+\omega$ $K(X)$ is the ...
4
votes
1answer
83 views

How to make an order isomorphism

Two linear orders $A$ and $B$ have starting points $a_0$ and $b_0$, and have cofinalities $\omega_1$. Let $(a_\alpha )_{\alpha<\omega_1}$ and $(b_\alpha )_{\alpha<\omega_1}$ be cofinal ...
10
votes
1answer
77 views

Martin's Axiom and products of c.c.c. spaces

A topological space $X$ has the c.c.c. if every family of pairwise disjoint non-empty open subsets of $X$ is countable. Consider the following statement: $(P)$ The product of two c.c.c. spaces is ...
2
votes
1answer
43 views

Question concerning a statement about separability

So here is my question, Let $X$ be topological space. If every disjoint familiy of open sets is at most countable, then $X$ is separable. I am pretty sure that this is true but I still wanted to ask ...
3
votes
1answer
92 views

Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
2
votes
1answer
31 views

Equivalent definitions of P-Points (Ultrafilters)

It is stated in lots of places (including this one) that these two are equivalent definitons for an ultrafilter $u$ to be a p-point. If for every sequence $\left < A_n: n \in \omega \right>$ ...
4
votes
2answers
64 views

Different definitions of P-Points (ultrafilters)

Recently I have been exposed to the concept of P-Points. I have three definitions: A non-principal ultrafilter $u$ is a P-Point if: For every sequence $\left < A_n \right >_{n\in \omega}$ of ...
0
votes
0answers
48 views

A special filter on cartesian product of sets

The following is inspired by this article in nLab (in attempt to simplify it using my notions of funcoids and reloids, which notation is however outside of the scope of this question). Fix a set $U$. ...
2
votes
2answers
98 views

What is the cardinality of the open ordinal space $[0,\Omega)$ if we remove it's limit ordinals?

Let $\Omega$ be the first uncountable ordinal. and for any limit ordinal $\lambda < \Omega$ Let $U_\lambda$ an open neighborhood of $\lambda$ with the order topology. what is the cardinality of the ...
1
vote
1answer
104 views

Is any subset of the open ordinal space $[0,\Omega)$ $G_\delta$?

Consider the open ordinal space $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal. Can I say that every subset of $[0,\Omega)$ is $G_\delta$? If yes, does this imply that $[0,\Omega)$ is ...
5
votes
1answer
177 views

Compact metric spaces is second countable and axiom of countable choice

Why we need axiom of countable choice to prove following theorem: every compact metric spaces is second countable? In which step it's "hidden"? Thank you for any help.
6
votes
1answer
176 views

Can we prove that every ordered space is normal without choice?

In ZFC, every linear ordered space respect to the order topology is completely normal. I saw the this proof and proof of this statement in the book "Counterexamples of topology" (Example 39). But as I ...
5
votes
1answer
122 views

Is $\{0,1\}^{\omega_1}$ sequentially compact?

It is claimed that an uncountable product of $[0,1]$ is not sequentially compact, e.g. in Wikipedia (I think replacing $[0,1]$ by $\{0,1\}$ doesn't make much difference). However, the constructions I ...
2
votes
0answers
182 views

An exercise from the handbook of set-theoretic topology

This is an exercise from the handbook of set-theoretic topology (Exercise 13.3): Assume $\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact ...
3
votes
1answer
257 views

Cardinality of set of discontinuities of cadlag functions

I know that non-decreasing cadlag functions (functions that are right continuous with left limits) on $[0,\infty)$ have at most a countable number of discontinuities. Does the same result hold for ...
4
votes
1answer
116 views

Question regarding disjoint unions, sequential compactness, and Dedekind-finiteness

I have proved the following two results: $[\mathsf{ZF}]$ The disjoint union of a Dedekind-finite family of sequentially compact topological spaces is again sequentially compact. ...
7
votes
1answer
58 views

Existence of a specific type of ultrafilter

Does there exist an ultrafilter $\omega$ on $N$ with the following property: $\forall A \in \omega$ we have that $kA \in \omega$ for a fixed $k \in N$. Where $kA = \{ka : a \in A\}$. I do not ...
1
vote
0answers
74 views

$U,V\subseteq X^\omega$ and X is w.o. such that $U'\subseteq U$ and $V'\subseteq V$ open and $U'\cap V'=\emptyset$ and $U'\cup V'=U\cup V$

Suppose $U,V\subseteq X^\omega$ (where $X^\omega:=\{f| f:\omega\to X\}$) and X is wellorderable, how it would be possible to show that there are open sets $U'\subseteq U$ and $V'\subseteq V$ such that ...
1
vote
1answer
101 views

Show that $X$ is separable.

I'm working through Kunen's Set Theory and I'm not sure how to proceed on part of one exercise. Let $X$ be compact Hausdorff and $\mathbb{O}_X$ be the poset of nonempty open sets of $X$ ordered by ...
6
votes
1answer
129 views

Classifying “nice” topological spaces on (occasionally pathological) sets

Background: By a unique convergence (UC) topology, I mean a topology under which sequences converge to at most one point. Suppose we are given a set $X$ and a topology $\mathcal T$ on $X$. It can be ...
1
vote
1answer
104 views

What is a finite partial function from $\mathfrak c$ to $D=\{0,1\}$

While I reading a paper, there is a notation is called finite partial function. I searched by google, I cannot find its definition. So I post it here as a question: What is a finite partial ...
1
vote
1answer
74 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
1answer
49 views

Cardinality of Hausdorff Space

Here is a theorem which proof I don't understand (taken from R. Engelking, General Topology). Theorem: For every Hausdorff space $X$ we have $|X| \le \operatorname{exp}\operatorname{exp} d(X)$. ...
9
votes
2answers
307 views

weak* separable question

(In another question Nate Eldredge said I should ask this.) Let $X$ be a Banach space, $X^\ast$ the dual space, and $B_{X^\ast}$ the unit ball of $X^\ast$. In the weak* topology for $X^\ast$, does ...
1
vote
1answer
109 views

How many Borel conjectures are there

The following may be referred to as Borel conjecture: Every strong measure zero set of reals is countable. On the other hand Wikipedia refers to the following as the Borel conjecture: Let $M$ and ...
4
votes
0answers
77 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 $\{H_\alpha: \alpha < \kappa\}$ ...
8
votes
2answers
189 views

A possibly set theoretic question in general topology

Let $A\subseteq \scr{P}$$(X)$$,B\subseteq$$\scr{P}$$( Y)$. Let $\tau_X$ be the smallest topology on $X$ that contains the set $A$. Let $\tau_Y$ be the smallest topology on $Y$ that contains the set ...
8
votes
1answer
463 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 ...
25
votes
1answer
612 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 ...
10
votes
0answers
168 views

Restrictions of null/meager ideal

Let I denote the null/meager ideal on reals. Is it consistent that for any pair of non null/meager sets A and B, there is a null/meager preserving bijection between A and B? In particular, is this ...
1
vote
2answers
98 views

On the small first countable regular spaces

Here are some definitions. A countably infinite closed discrete set $A \subset X$ has property $D$ in $X$ provided there exists a discrete family of open sets $\{U_a : a \in A\}$ ...
3
votes
1answer
60 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
147 views

well-ordering principle

I'm confused by the well-ordering principle . The proof is clear but I don't have any idea why is it true . It says that every non-empty set can be well-ordered but $C^{1}$ is a non-empty set but ...
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 ...
5
votes
1answer
107 views

Why do the cardinal functions exist in topological history?

Why do the cardinal functions exist in topological history? In other words, Why are they useful? Thanks ahead.
5
votes
1answer
78 views

First countable, ccc, Hausdorff space

How to prove that every first countable, ccc, Hausdorff space has cardinality at most $2^\omega$ by use the Erdős-Rado theorem? Erdős-Rado theorem: Let $\kappa$ be an infinite cardinal. Let $E$ be ...
7
votes
1answer
154 views

Set theoretic implications of constructions in Differential Geometry/ Topology

In subjects like Differential Geometry/ General Topology one often constructs for each $x$ in a space $X$ a set $U_x$ satisfying certain properties. Examples where one does constructions like this: ...
4
votes
1answer
93 views

Why is a p-point never the limit of a countable set of non-principal ultrafilters?

There are several descriptions, but I think the following definition of p-point suits for now: A point such that the intersection of countably many neighbourhoods of it, is again a neighbourhood. ...
2
votes
1answer
62 views

Is there an interesting topology on the transfinite cardinals?

Presumably there will be a problem talking about all the transfinite cardinals, but assuming that this can be got around; is there an interesting topology on it - say such that the limit cardinals are ...
1
vote
2answers
65 views

A question on a Luzin space

A Luzin space is a crowded (no isolated points) Hausdorff space in which every nowhere dense set is countable. Now there is a Luzin space with its cardinality is $\omega_1$. Is it a Lindelöf space? ...
4
votes
1answer
72 views

A question on $\omega_1$

For the liner order space $\omega_1$, does it have countable chain condition? (I kown it is not separable, because for any countable subset $A$ of $\omega_1$, there exists an ordinal $\kappa$ ...
1
vote
1answer
93 views

How big such set is?

Let $A \subset R$, where $R$ is the real line. Let $F=\{f\mid f: A\rightarrow R^\omega \text{ is continuous}\}$. How big is the set $F$? Thanks for your help.
0
votes
1answer
35 views

What is Fin(E,2)?

It is easy to see that Bor(I) satisfies the c.c.c., because the measures add up to at most 1. Fin(E,2) is also c.c.c., but the proof is more difficult. (Countable chain condition, wikipedia) So ...
1
vote
1answer
40 views

A question about linearly lindelöf space

Suppose $X$ be a linearly lindelöf space, and $\mathcal{P}$ a family closed subsets of $X$, satisfying $\cap \mathcal{P}=\emptyset$. Let $\tau$ be the smallest cardinal number such that there is a ...
1
vote
1answer
58 views

Is every Hausdorff sequential space monolithic?

A space $X$ called monolithic if for any subset $A$ with $|A|\le \kappa$, then $nw(\overline{A})\le \kappa$. Is every Hausdorff sequential space monolithic? What I have tried: For any subset $A$, ...