User paul - MathOverflow

the example of ccc but not separable

2011-10-20T03:58:52Z

I am interested in the relation between the property of countable chain condition (ccc) and the property of separable. Could someone recommend some papers or books about this to me? thanks in advance.

A conjecture on closed discrete subset

2013-05-04T03:04:34Z

I am struggling with this old problem, which is also posted here:

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 $\mathfrak c$.

$X$ has a regular $G_\delta$-diagonal iff there is a collection of open sets of $X^2$, say $\lbrace U_n: n\in N\rbrace$, such that $\Delta=\bigcap\lbrace \overline{U_n}: n \in N\rbrace$, where $\Delta=\lbrace(x,x): x \in X\rbrace$.

Note that the question is answered; however I hope to get new proof.

By certain effort, If $|X|>\mathfrak c$, we can get an uncountable closed discrete subset $S$ of $X$, and for any point $x \in X$, there exists an open set $U_x$ such that $\overline{U_x} \cap S$ has at most one point.

I would like to know whether the following conjecture is right, wrong, or neither:

Let $X$ be a Hausdorff space. If $S \subset X$ is an uncountable closed discrete subset of $X$, and for any point $x \in X$, there exists an open set $U_x$ such that $\overline{U_x} \cap S$ has at most one point. Then could we obtain an uncountale collection of disjoint open sets in $X$?

Thanks for your any help.

A question on continuous mappings

2013-05-04T02:58:25Z

The question is also posted here.

Let $M=\mathbb{R}$ and $\tau_M=\lbrace U\cup A: U$ open in $\mathbb{R}, A\subset \mathbb{R} \setminus B\rbrace$, where $B$ is a Bernstein set. Then $(M,\tau_M)$ is a topological space called the Michael Line. It is a regular Lindelof space.

Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space.

Let $f: M \to X$ be any one-to-one and onto continuous mapping. Then is $X$ always submetrizable?

Thanks for your help.

A question on star $\sigma$-compact spaces

2013-05-01T10:59:33Z

The question is also posted here.

A topological space $X$ is said to be star $\sigma$-compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a $\sigma$-compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$.

Let $X$ have countable extent and is locally compact. Then must $X^2$ be star $\sigma$-compact?

Thanks for your help.

A question on metrizable space

2013-02-21T12:11:02Z

Q1, Does a metrizable space $X$ with $e(X)=\omega$ (i.e., it has countable extent) which is not lindelof exist?

Q2, Let $X$ be the one point lindefication of a discret space of cardinality $\omega_1$ and $Y$ is any Lindelof space. Is $X \times Y$ always Lindelof?

Thanks for any help.

Is $f$ continuous?

2013-04-26T07:06:47Z

The question is also posted here.

The paper is Mizokami : On characterizations of spaces with $G_\delta$-diagonals

See its Theorem 1, also you can see the picture . http://picpaste.com/a-eaiF4d3t.bmp.

Theorem 1: A space $X$ has a $G_\delta$-diagonal iff there is an open mapping (single valued) $f$ from a metric space $T$ onto $X$ such that $$d(f^{-1}(p),f^{-1}(q))>0,$$ for distinct points $p, q \in X.$

The author difines $T$ as follows:

$T=\lbrace (\alpha_1,\alpha_2,...)\in N(A): \bigcap \lbrace U_{\alpha_n}^n: n\in N\rbrace\not=\emptyset \rbrace$, where $\lbrace \mathcal U_n=\lbrace U_{\alpha}^n: \alpha \in A, n \in N\rbrace$ is a sequence of open covering of $X$ satisfying the condition in Lemma 1. (it can be seen in the paper.)

The author difines $f: T \rightarrow X$ as follows:

$f(\alpha)=\bigcap \lbrace U_{\alpha_n}^n: n\in N \rbrace$ for $\alpha \in T$

My question is this:

1) What is the topology the author used which make $T$ is metrizable?

2) Is $f$ continuous?

Thanks for your help.

Answer by Paul for What is the smallest cardinality a topology can have which is c.c.c but not separable (in ZFC)?

2013-03-29T02:51:08Z

This may be helpful for you:

Let $X$ be a space with $|X|= \aleph_1$, let $\tau_X= \lbrace U: X\setminus U \text{ is countable } \rbrace$. This space is CCC, but not separable.

Proof: $X$ is not separable: for any countable set $A \subset X$, clearly, $U=X\setminus A$ is open and $U \cap A=\emptyset$.

$X$ is CCC: if $X$ has uncountable disjoint open sets $\lbrace \cal U_\xi: \xi \in \aleph_1\rbrace$. Pick one open set, for example, $U_0$. Because $X \setminus U_0$ is uncountable, it is a contradiction with $U_0$ is open.

A question on countably compact space

2013-03-17T12:35:56Z

A regular space $X$ is

star compact (which implies pseudocompact)
with $G_\delta$-diagonal
star countable
first countable
$e(X)\le \aleph_0$ ( in fact it implies star countable)
$|X|=\aleph_1$
Cech-complete
under CH

My question is this: Must $X$ be countably compact?

Thanks ahead.

A question on linearly lindelof space

2013-03-07T00:32:47Z

Let $X$ is a linearly lindelof subspace of $Z$ and $b$ is not $\omega$-separated from $X$, i.e., for any closed $G_\delta$ set $P$ of $Z$ which contains $b$, $P\cap X \not=\emptyset$. If $\tau < \aleph_\omega$, how to show that $b$ is not $\tau$-separated from $X$, i.e., for any closed $G_\tau$ set $P$ of $Z$ which contains $b$, $P\cap X \not=\emptyset$? Thanks ahead:)

A question from Arhangel'skii-Buzyakova

2013-03-06T03:10:41Z

The question is also posted here, however there is no answer.

Recently, I am reading the paper: On linearly Lindelöf and strongly discretely Lindelöf spaces by Arhangel'skii and Buzyakova. Here is the Lemma 2.2 in paper. (Sorry for the picture is not clear.)

The fifth line from last. How could I see that for any $a\in H$ and $z\in Z\setminus H$, there exists an element $V$ of $\mathcal{U}$ such that $a\in V$ and $z\notin V$? Thanks very much.

Why $z \in \overline{A}$?

2013-02-25T13:10:50Z

In the Picture blew: The paper can be downloaded here. Why $z \in \overline{A}$? Thanks.

A point $x$ of a space $X$ is called $G_\omega$-separated from a subset $Y$ of $X$ if there is a closed $G_\omega$ -set $P$ in $X$ such that $x \in P$ and the sets $Y$ and $P$ are disjoint.

How to see such space is Lindelof?

2013-02-20T08:00:52Z

Let $R$ denote the set of all real numbers. $B$ is any Bernstein set of $R$.

Bernstein Set: A subset of the real line that meets every uncountable closed subset of the real line but that contains none of them. It's from wiki.

We topologize $R$ now: the set $B$ is discrete and its complement has the usual topology. How to see the new topological space is Lindelof?

Is the Sorgenfrey Line monotonically monolithic?

2013-02-19T11:31:38Z

Just as the title explains, is the Sorgenfrey Line monotonically monolithic (see the definition)?

Does the network of $X$ equal to the network of $C_p(X)$?

2013-02-08T12:36:47Z

Does the network of $X$ equal to the network of $C_p(X)$?

$C_p(X)$ denotes the set of all real-valued continuous functions on $X$ endowed with the topology of pointwise convergence.

Thanks!

An open problem on general topology

2013-02-06T00:47:58Z

There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.

Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a $G_\delta$-diagonal?

A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a compact subspace $K$ of $X$ such that $X = \operatorname{St}(K,\mathscr{U})$. More on star compactness see here.

My question is this: Is always the cardinality of such regular (Tychonoff) star compact space less than $2^{\omega_0}$? See the related link here.

A question on hereditary Lindelof number

2013-01-19T11:22:08Z

Let $X$ be space. A space $X$ is called right-separated if it can be well-ordered in such a way that every initial segment is open in $X$. See the related link (left-separated).

How could we show that hereditary lindelof number is the supremum of cardinalities of right-separated subspaces of $X$?

Understanding the left-separated spaces

2013-01-14T06:35:45Z

A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$.

Could someone post some left-separated space to help me understand such definition?

a stationary set of successor cardinal

2012-01-14T10:03:46Z

I don't know how to proof a question which I meet in a textbook. It is this: Let $\alpha$ be a successor cardinal, and $S\subset \alpha$ be a statinonary set, then $S$ can be seen as the union of $\alpha$ disjoint stationary sets?

Could someone give me the key points to solve the question?

Any help wil be appreciated. Thanks ahead:)

Could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology?

2012-01-06T00:47:25Z

Let $X$ be a Tychonof space and $\beta X$ is its compactification. Then could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology?

metrizable space

2012-01-05T11:35:43Z

Suppose $X=\cup K_n$ is a topological space, $K_n$ is a metrizable subspace in $X$ for every $n \in \omega$, then $X$ is a metrizable space?

In metrizable spaces, compactness is equivalent to $\sigma$-compactness?

One more: Is pseudocompactness hereditary with respect to $\sigma$-compact subspace?

some questions on Lindelöf property

2012-01-02T09:37:03Z

I have several questions on Lindelöf property.

If every point countable open cover of $X$ has a countable subcover (Condition A), does $X$ have Lindelöf property? How far is having Condition A from Lindelöf property?

A space $X$ is called $\omega_1$-Lindelöf if every $\omega_1$-sized open cover of $X$ contains a countable subcover.

Can every $\omega_1$-Lindelöf space with Condition A be Lindelöf?

A space $X$ is called discretely Lindelöf if the closure of every discrete subspace of $X$ is Lindelöf.

Can every discretely Lindelöf space with Condition A be Lindelöf?

Does X have any diagonal properties?

2011-11-28T02:26:57Z

Assume that $2^{\omega_1}=2^\omega=\mathfrak{c}$. Let $D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$={$\xi<\mathfrak{c}:y(\xi)=1$}, the support of $y$, and let $X$={$x\in Y:0<|\operatorname{supp}(x)|\le\omega_1$}; $|X|=\mathfrak{c}^{\omega_1}=(2^{\omega_1})^{\omega_1}=\mathfrak{c}$.

Does X have $G_\delta$ diagonal?

If a topological space X has $\aleph_1$-calibre, then it must be star countable?

2011-11-14T12:55:30Z

If a topological space X has $\aleph_1$-calibre[definition], then it must be star countable? What if the cardinality of the topological space X is additionally < = $2^{\aleph_0}$?

on $F_\sigma$-discrete space

2011-11-09T12:47:23Z

A space is $F_\sigma$-discrete space if it is the countable union of closed discrete subspaces. Is it true that every subset of an $F_\sigma$-discrete space is of the type $G_\delta$?

Is there a countable pseudocharacter Hausdorff space，such that...?

2011-10-27T02:17:57Z

Let X be a Hausdorff space and Difine the Property A as following: if $\mathscr{U}$ is a collection of open sets of X that witnesses Hausdorff property of X (= $\forall x,y \in X$, there exist two disjoint opensets $U_1$ and $U_2$ $\in \mathscr{U}$, st $x \in U_1$ and $y \in U_2$ ), then there is a point $x\in X$ such that $|U \in \mathscr{U}: x \in U|>\omega$. Is there a countable pseudocharacter Hausdorff space X with the property A?

CCC +　collectionwise normality =>　paracompact?

2011-10-24T13:01:34Z

Is there a CCC and collectionwise normal space, that isn't paracompact?

As we know, CCC + monotone　normality => lindelof.

CCC +　collectionwise normality =>　paracompact?

CCC = countable chain condition

Collectionwise normality = if X is a $T_{1}$ space and for every discrete family

$\{F_{s}\}_{s \in S}$ of closed subsets of X there exists a discrete family

$\{V_{s}\}_{s \in S}$ of open subsets of X such that $F_{s}$ $\subset$ $V_{s}$ for every s

$\in$ S.

continuous function

2011-10-20T03:21:07Z

Suppose the countable subspace $D$ is dense in the separable Tychonoff space $X$ and $f$ is a continous function from $D$ to the closed unit interval. What are some conditions on $X$ or $D$, which make $f$ continuously extendable over $X$？

$G_\delta$-diagonal

2011-10-20T10:44:54Z

Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't

a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence

${G_n}$ of open sets belonging to $X^2$ with the diagonal $\Delta$ = $\cap{G_n}$.

zeroset-diagonal

2011-10-18T01:17:04Z

Is it true that a topology space X with a zeroset diagonal is first countable?

what if X is additionally CCC?

$\aleph_1$-calibre

2011-10-18T02:49:48Z

The square of X which is $\aleph_1$-calibre is still $\aleph_1$-calibre?

Comment by Paul

2013-05-04T11:27:21Z

what is meaning of non-measurable cardinality?

Comment by Paul

2013-05-04T08:43:03Z

Buzyakova posted a new definition of absolutely submetrizable (= every Tychonoff subtopology is submetrizable) in the paper: On absolutely submetrizable spaces

Comment by Paul

2013-04-26T07:42:52Z

Yes. He may be very old.

Comment by Paul

2013-04-26T07:16:57Z

The author said unclearly in the paper.

Comment by Paul

2013-03-18T02:10:20Z

@Ali: could you give me a link of the book, so I can download it?

Comment by Paul

2013-03-18T01:45:01Z

It may be not at all.

Comment by Paul

2013-03-18T01:42:50Z

What is meaning of "5l"?

Comment by Paul

2013-03-07T00:07:21Z

Thanks for the answer. So it is wrong at $x\in Z\setminus H$, where it should be $x\in X\setminus H$. Am I right?

Comment by Paul

2013-02-26T04:11:20Z

@Todd Eisworth: Yes. I got it. Thanks!

Comment by Paul

2013-02-26T03:50:10Z

@Todd Eisworth: How could you see that $\cap V_i$ is closed as a closed $G_\omega$-set $P$ of $z$?

Comment by Paul

2013-02-26T03:45:15Z

@Yemon Choi: You are not very welcome. It seems that you only can do this, what other could you do?

Comment by Paul

2013-02-25T01:28:34Z

If we take $A=X$, of courese, for any $B \subset A$, we have $\overline{B}\subset A$. It is nonsense.

Comment by Paul

2013-02-22T03:19:04Z

@Martin: Yes. thanks. I get it.

Comment by Paul

2013-02-22T02:13:39Z

Sorry. What is your meaing? Could you express clearly?

Comment by Paul

2013-02-22T00:23:14Z

The mathod very good!

User paul - MathOverflow

the example of ccc but not separable

A conjecture on closed discrete subset

A question on continuous mappings

A question on star $\sigma$-compact spaces

A question on metrizable space

Is $f$ continuous?

Answer by Paul for What is the smallest cardinality a topology can have which is c.c.c but not separable (in ZFC)?

A question on countably compact space

A question on linearly lindelof space

A question from Arhangel'skii-Buzyakova

Why $z \in \overline{A}$?

How to see such space is Lindelof?

Is the Sorgenfrey Line monotonically monolithic?

Does the network of $X$ equal to the network of $C_p(X)$?

An open problem on general topology

A question on hereditary Lindelof number

Understanding the left-separated spaces

a stationary set of successor cardinal

Could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology?

metrizable space

some questions on Lindelöf property

Does X have any diagonal properties?

If a topological space X has $\aleph_1$-calibre, then it must be star countable?

on $F_\sigma$-discrete space

Is there a countable pseudocharacter Hausdorff space，such that...?

CCC + collectionwise normality => paracompact?

continuous function

$G_\delta$-diagonal

zeroset-diagonal

$\aleph_1$-calibre

Comment by Paul

Comment by Paul

Comment by Paul

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CCC +　collectionwise normality =>　paracompact?