# Tagged Questions

58 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 ...
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 ...
82 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 ...
71 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 ...
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 ...
86 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 ...
28 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 $\left < A_n: n \in \omega \right>$ ...
60 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 ...
41 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 $U$. ...
95 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 ...
91 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 ...
155 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.
159 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 ...
118 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 ...
176 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 ...
238 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 ...
113 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. ...
57 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 ...
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 ...
99 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 ...
121 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 ...
94 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 ...
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 ...
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 $|X| \le \operatorname{exp}\operatorname{exp} d(X)$. ...
284 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 ...
103 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 ...
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 $\{H_\alpha: \alpha < \kappa\}$ ...
188 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 ...
445 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 ...
611 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 ...
159 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 ...
96 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 $\{U_a : a \in A\}$ ...
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 ...
136 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 ...
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 ...
105 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.
77 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 ...
152 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: ...
87 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. ...
61 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 ...
64 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? ...
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 $\kappa$ ...
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.
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 ...
39 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 ...
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 $A$, ...
101 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 $A$ ...
126 views

### Does every metrizable space have point-countable base?

Just as the title explains: Does every metrizable space have point-countable base? Thanks ahead:)