User paul - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:13:10Z http://mathoverflow.net/feeds/user/18465 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78641/the-example-of-ccc-but-not-separable the example of ccc but not separable Paul 2011-10-20T03:58:52Z 2013-05-15T12:53:50Z <p>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.</p> http://mathoverflow.net/questions/129605/a-conjecture-on-closed-discrete-subset A conjecture on closed discrete subset Paul 2013-05-04T03:04:34Z 2013-05-06T07:24:54Z <p>I am struggling with this old problem, which is also posted <a href="http://math.stackexchange.com/questions/380022/a-conjecture-on-closed-discrete-subset" rel="nofollow">here</a>:</p> <blockquote> <p>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 $\mathfrak c$.</p> </blockquote> <p><strong>$X$ has a regular $G_\delta$-diagonal</strong> 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 $\Delta=\lbrace(x,x): x \in X\rbrace$.</p> <p>Note that the question is answered; however I hope to get new proof. </p> <p>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.</p> <p>I would like to know whether the following conjecture is right, wrong, or neither:</p> <blockquote> <p>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 $X$?</p> </blockquote> <p>Thanks for your any help.</p> http://mathoverflow.net/questions/129604/a-question-on-continuous-mappings A question on continuous mappings Paul 2013-05-04T02:58:25Z 2013-05-04T02:58:25Z <p>The question is also posted <a href="http://math.stackexchange.com/questions/377990/a-question-on-continuous-mappings" rel="nofollow">here</a>.</p> <p>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 <a href="http://math.stackexchange.com/questions/377990/a-question-on-continuous-mappings" rel="nofollow">Bernstein set</a>. Then $(M,\tau_M)$ is a topological space called the <strong>Michael Line</strong>. It is a regular Lindelof space.</p> <p><strong>Submetrizable</strong> = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space.</p> <blockquote> <p>Let $f: M \to X$ be any one-to-one and onto continuous mapping. Then is $X$ always submetrizable?</p> </blockquote> <p>Thanks for your help.</p> http://mathoverflow.net/questions/129297/a-question-on-star-sigma-compact-spaces A question on star $\sigma$-compact spaces Paul 2013-05-01T10:59:33Z 2013-05-01T10:59:33Z <p>The question is also posted <a href="http://math.stackexchange.com/questions/376843/a-question-on-star-sigma-compact-spaces" rel="nofollow">here</a>.</p> <p>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 $X = \operatorname{St}(K,\mathscr{U})$.</p> <p>Let $X$ have countable extent and is locally compact. Then must $X^2$ be star $\sigma$-compact?</p> <p>Thanks for your help.</p> http://mathoverflow.net/questions/122528/a-question-on-metrizable-space A question on metrizable space Paul 2013-02-21T12:11:02Z 2013-04-30T04:29:08Z <p>Q1, Does a metrizable space $X$ with $e(X)=\omega$ (i.e., it has countable extent) which is not lindelof exist? </p> <p>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?</p> <p>Thanks for any help. </p> http://mathoverflow.net/questions/128806/is-f-continuous Is $f$ continuous? Paul 2013-04-26T07:06:47Z 2013-04-26T12:14:34Z <p>The question is also posted <a href="http://math.stackexchange.com/questions/373037/what-is-the-topology-the-author-used-which-make-t-is-metrizable" rel="nofollow">here.</a></p> <p>The paper is Mizokami : <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.pja/1195518845" rel="nofollow">On characterizations of spaces with $G_\delta$-diagonals</a></p> <p>See its Theorem 1, also you can see the picture . <a href="http://picpaste.com/a-eaiF4d3t.bmp" rel="nofollow">http://picpaste.com/a-eaiF4d3t.bmp</a>.</p> <blockquote> <p><strong>Theorem 1:</strong> 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 $p, q \in X.$</p> </blockquote> <p>The author difines $T$ as follows:</p> <blockquote> <p><strong>$T=\lbrace (\alpha_1,\alpha_2,...)\in N(A): \bigcap \lbrace U_{\alpha_n}^n: n\in N\rbrace\not=\emptyset \rbrace$, 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.)</strong></p> </blockquote> <p>The author difines $f: T \rightarrow X$ as follows:</p> <blockquote> <p><strong>$f(\alpha)=\bigcap \lbrace U_{\alpha_n}^n: n\in N \rbrace$ for $\alpha \in T$</strong></p> </blockquote> <p>My question is this:</p> <p>1) <strong>What is the topology the author used which make $T$ is metrizable?</strong></p> <p>2) <strong>Is $f$ continuous?</strong></p> <p>Thanks for your help.</p> http://mathoverflow.net/questions/125727/what-is-the-smallest-cardinality-a-topology-can-have-which-is-c-c-c-but-not-separ/125869#125869 Answer by Paul for What is the smallest cardinality a topology can have which is c.c.c but not separable (in ZFC)? Paul 2013-03-29T02:51:08Z 2013-03-29T05:14:24Z <p>This may be helpful for you:</p> <blockquote> <p>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.</p> </blockquote> <p>Proof: <strong>$X$ is not separable:</strong> for any countable set $A \subset X$, clearly, $U=X\setminus A$ is open and $U \cap A=\emptyset$.</p> <p><strong>$X$ is CCC:</strong> 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.</p> http://mathoverflow.net/questions/124778/a-question-on-countably-compact-space A question on countably compact space Paul 2013-03-17T12:35:56Z 2013-03-18T01:41:09Z <p>A regular space $X$ is </p> <ol> <li>star compact (which implies pseudocompact)</li> <li>with $G_\delta$-diagonal</li> <li>star countable</li> <li>first countable</li> <li>$e(X)\le \aleph_0$ ( in fact it implies star countable)</li> <li>$|X|=\aleph_1$</li> <li>Cech-complete</li> <li>under CH</li> </ol> <p>My question is this: Must $X$ be countably compact?</p> <p>Thanks ahead.</p> http://mathoverflow.net/questions/123828/a-question-on-linearly-lindelof-space A question on linearly lindelof space Paul 2013-03-07T00:32:47Z 2013-03-07T19:12:27Z <p>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 $\tau &lt; \aleph_\omega$, 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:)</p> http://mathoverflow.net/questions/123697/a-question-from-arhangelskii-buzyakova A question from Arhangel'skii-Buzyakova Paul 2013-03-06T03:10:41Z 2013-03-06T20:27:14Z <p>The question is also posted <a href="http://math.stackexchange.com/questions/320334/a-question-from-arhangelskii-buzyakova" rel="nofollow">here</a>, however there is no answer.</p> <p>Recently, I am reading the paper: <a href="http://www.ams.org/journals/proc/1999-127-08/S0002-9939-99-04783-8/" rel="nofollow"><em>On linearly Lindelöf and strongly discretely Lindelöf spaces</em> by Arhangel'skii and Buzyakova.</a> Here is the Lemma 2.2 in paper. (Sorry for the picture is not clear.)</p> <p><img src="http://i.stack.imgur.com/ML3UU.png" alt="enter image description here"></p> <p>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.</p> http://mathoverflow.net/questions/122878/why-z-in-overlinea Why $z \in \overline{A}$? Paul 2013-02-25T13:10:50Z 2013-02-25T15:54:14Z <p>In the Picture blew:<img src="http://i.stack.imgur.com/JqZ9z.png" alt="enter image description here"> The paper can be downloaded <a href="http://www.ams.org/journals/proc/1999-127-08/S0002-9939-99-04783-8/" rel="nofollow">here</a>. Why $z \in \overline{A}$? Thanks.</p> <p>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.</p> http://mathoverflow.net/questions/122382/how-to-see-such-space-is-lindelof How to see such space is Lindelof? Paul 2013-02-20T08:00:52Z 2013-02-20T08:31:43Z <p>Let $R$ denote the set of all real numbers. $B$ is any Bernstein set of $R$. </p> <blockquote> <p>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.</p> </blockquote> <p>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?</p> http://mathoverflow.net/questions/122288/is-the-sorgenfrey-line-monotonically-monolithic Is the Sorgenfrey Line monotonically monolithic? Paul 2013-02-19T11:31:38Z 2013-02-19T12:32:47Z <p>Just as the title explains, is the Sorgenfrey Line monotonically monolithic <a href="http://math.stackexchange.com/questions/297948/what-is-the-relation-between-kappa-monolithic-and-monotonically-monolithic" rel="nofollow">(see the definition)</a>?</p> http://mathoverflow.net/questions/121182/does-the-network-of-x-equal-to-the-network-of-c-px Does the network of $X$ equal to the network of $C_p(X)$? Paul 2013-02-08T12:36:47Z 2013-02-08T18:34:22Z <p>Does the network of $X$ equal to the network of $C_p(X)$?</p> <p>$C_p(X)$ denotes the set of all real-valued continuous functions on $X$ endowed with the topology of pointwise convergence.</p> <p>Thanks!</p> http://mathoverflow.net/questions/120922/an-open-problem-on-general-topology An open problem on general topology Paul 2013-02-06T00:47:58Z 2013-02-06T06:25:48Z <p>There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.</p> <p><strong>Problem 4.8.</strong> Is a regular (Tychonoff) star compact space metrizable if it has a $G_\delta$-diagonal?</p> <p>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 <a href="http://math.stackexchange.com/questions/94172/how-far-is-being-star-compact-from-being-countably-compact" rel="nofollow">here</a>.</p> <p>My question is this: Is always the cardinality of such regular (Tychonoff) star compact space less than $2^{\omega_0}$? See the related link <a href="http://math.stackexchange.com/questions/295355/an-open-problem-on-general-topology" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/119319/a-question-on-hereditary-lindelof-number A question on hereditary Lindelof number Paul 2013-01-19T11:22:08Z 2013-01-19T20:31:27Z <p>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 $X$. <a href="http://mathoverflow.net/questions/118858/understanding-the-left-separated-spaces" rel="nofollow">See the related link (left-separated).</a></p> <p>How could we show that hereditary lindelof number is the supremum of cardinalities of right-separated subspaces of $X$?</p> http://mathoverflow.net/questions/118858/understanding-the-left-separated-spaces Understanding the left-separated spaces Paul 2013-01-14T06:35:45Z 2013-01-14T13:51:21Z <p><strong>A space $X$ is called left-separated if it can be well-ordered in such a way that every initial segment is closed in $X$.</strong></p> <p>Could someone post some left-separated space to help me understand such definition?</p> http://mathoverflow.net/questions/85646/a-stationary-set-of-successor-cardinal a stationary set of successor cardinal Paul 2012-01-14T10:03:46Z 2012-01-14T13:02:59Z <p>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? </p> <p>Could someone give me the key points to solve the question? </p> <p>Any help wil be appreciated. Thanks ahead:)</p> http://mathoverflow.net/questions/85017/could-ix-be-seen-as-a-subspace-of-i-beta-x-under-the-compact-open-topolog Could $I^X$ be seen as a subspace of $I^{\beta X}$ under the compact-open topology? Paul 2012-01-06T00:47:25Z 2012-01-10T07:30:30Z <p>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?</p> http://mathoverflow.net/questions/84950/metrizable-space metrizable space Paul 2012-01-05T11:35:43Z 2012-01-05T19:25:37Z <p>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?</p> <p>In metrizable spaces, compactness is equivalent to $\sigma$-compactness?</p> <p><strong>One more:</strong> Is pseudocompactness hereditary with respect to $\sigma$-compact subspace?</p> http://mathoverflow.net/questions/84726/some-questions-on-lindelof-property some questions on Lindelöf property Paul 2012-01-02T09:37:03Z 2012-01-04T21:45:03Z <p>I have several questions on Lindelöf property.</p> <p>If every point countable open cover of $X$ has a countable subcover (<strong>Condition A</strong>), does $X$ have Lindelöf property? How far is having <strong>Condition A</strong> from Lindelöf property?</p> <p><strong>A space $X$ is called $\omega_1$-Lindelöf if every $\omega_1$-sized open cover of $X$ contains a countable subcover.</strong></p> <p>Can every $\omega_1$-Lindelöf space with <strong>Condition A</strong> be Lindelöf?</p> <p><strong>A space $X$ is called discretely Lindelöf if the closure of every discrete subspace of $X$ is Lindelöf.</strong></p> <p>Can every discretely Lindelöf space with <strong>Condition A</strong> be Lindelöf?</p> http://mathoverflow.net/questions/82039/does-x-have-any-diagonal-properties Does X have any diagonal properties? Paul 2011-11-28T02:26:57Z 2011-11-29T12:40:26Z <p>Assume that $2^{\omega_1}=2^\omega=\mathfrak{c}$. Let $D$={ 0,1 }, and let $Y=D^\mathfrak{c}$. For $y\in Y\;$ let $\operatorname{supp}(y)$={$\xi&lt;\mathfrak{c}:y(\xi)=1$}, the <em>support</em> of $y$, and let $X$={$x\in Y:0&lt;|\operatorname{supp}(x)|\le\omega_1$}; $|X|=\mathfrak{c}^{\omega_1}=(2^{\omega_1})^{\omega_1}=\mathfrak{c}$. </p> <p>Does X have $G_\delta$ diagonal?</p> http://mathoverflow.net/questions/80890/if-a-topological-space-x-has-aleph-1-calibre-then-it-must-be-star-countable If a topological space X has $\aleph_1$-calibre, then it must be star countable? Paul 2011-11-14T12:55:30Z 2011-11-16T00:28:41Z <p>If a topological space X has $\aleph_1$-calibre<a href="http://mathoverflow.net/questions/78414/aleph-1-calibre/78451#78451" rel="nofollow">[definition]</a>, then it must be star countable? What if the cardinality of the topological space X is additionally &lt; = $2^{\aleph_0}$?</p> http://mathoverflow.net/questions/80480/on-f-sigma-discrete-space on $F_\sigma$-discrete space Paul 2011-11-09T12:47:23Z 2011-11-09T22:58:35Z <p>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 $G_\delta$?</p> http://mathoverflow.net/questions/79222/is-there-a-countable-pseudocharacter-hausdorff-spacesuch-that Is there a countable pseudocharacter Hausdorff space，such that...? Paul 2011-10-27T02:17:57Z 2011-10-28T05:01:48Z <p>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?</p> http://mathoverflow.net/questions/78975/ccc-collectionwise-normality-paracompact CCC +　collectionwise normality =>　paracompact? Paul 2011-10-24T13:01:34Z 2011-10-24T20:57:52Z <p>Is there a CCC and collectionwise normal space, that isn't paracompact? </p> <p>As we know, CCC + monotone　normality => lindelof.</p> <p>CCC +　collectionwise normality =>　paracompact?</p> <p>CCC = countable chain condition</p> <p>Collectionwise normality = if X is a <code>$T_{1}$</code> space and for every discrete family </p> <p><code>$\{F_{s}\}_{s \in S}$</code> of closed subsets of X there exists a discrete family </p> <p><code>$\{V_{s}\}_{s \in S}$</code> of open subsets of X such that <code>$F_{s}$ $\subset$</code> <code>$V_{s}$</code> for every s </p> <p>$\in$ S.</p> http://mathoverflow.net/questions/78637/continuous-function continuous function Paul 2011-10-20T03:21:07Z 2011-10-20T20:11:08Z <p>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 $X$？</p> http://mathoverflow.net/questions/78665/g-delta-diagonal $G_\delta$-diagonal Paul 2011-10-20T10:44:54Z 2011-10-20T13:28:02Z <p>Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't </p> <p>a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence </p> <p>${G_n}$ of open sets belonging to $X^2$ with the diagonal $\Delta$ = $\cap{G_n}$.</p> http://mathoverflow.net/questions/78406/zeroset-diagonal zeroset-diagonal Paul 2011-10-18T01:17:04Z 2011-10-19T19:30:12Z <p>Is it true that a topology space X with a zeroset diagonal is first countable? </p> <p>what if X is additionally CCC?</p> http://mathoverflow.net/questions/78414/aleph-1-calibre $\aleph_1$-calibre Paul 2011-10-18T02:49:48Z 2011-10-18T12:51:56Z <p>The square of X which is $\aleph_1$-calibre is still $\aleph_1$-calibre?</p> http://mathoverflow.net/questions/129605/a-conjecture-on-closed-discrete-subset/129632#129632 Comment by Paul Paul 2013-05-04T11:27:21Z 2013-05-04T11:27:21Z what is meaning of non-measurable cardinality? http://mathoverflow.net/questions/129604/a-question-on-continuous-mappings Comment by Paul Paul 2013-05-04T08:43:03Z 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 http://mathoverflow.net/questions/128806/is-f-continuous Comment by Paul Paul 2013-04-26T07:42:52Z 2013-04-26T07:42:52Z Yes. He may be very old. http://mathoverflow.net/questions/128806/is-f-continuous Comment by Paul Paul 2013-04-26T07:16:57Z 2013-04-26T07:16:57Z The author said unclearly in the paper. http://mathoverflow.net/questions/124778/a-question-on-countably-compact-space/124831#124831 Comment by Paul Paul 2013-03-18T02:10:20Z 2013-03-18T02:10:20Z @Ali: could you give me a link of the book, so I can download it? http://mathoverflow.net/questions/124778/a-question-on-countably-compact-space Comment by Paul Paul 2013-03-18T01:45:01Z 2013-03-18T01:45:01Z It may be not at all. http://mathoverflow.net/questions/124778/a-question-on-countably-compact-space/124831#124831 Comment by Paul Paul 2013-03-18T01:42:50Z 2013-03-18T01:42:50Z What is meaning of &quot;5l&quot;? http://mathoverflow.net/questions/123697/a-question-from-arhangelskii-buzyakova/123810#123810 Comment by Paul Paul 2013-03-07T00:07:21Z 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? http://mathoverflow.net/questions/122878/why-z-in-overlinea/122899#122899 Comment by Paul Paul 2013-02-26T04:11:20Z 2013-02-26T04:11:20Z @Todd Eisworth: Yes. I got it. Thanks! http://mathoverflow.net/questions/122878/why-z-in-overlinea/122899#122899 Comment by Paul Paul 2013-02-26T03:50:10Z 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 $z$? http://mathoverflow.net/questions/122878/why-z-in-overlinea Comment by Paul Paul 2013-02-26T03:45:15Z 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? http://mathoverflow.net/questions/122847/is-the-space-countably-compact Comment by Paul Paul 2013-02-25T01:28:34Z 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. http://mathoverflow.net/questions/122589/does-every-metrizable-space-have-point-countable-base Comment by Paul Paul 2013-02-22T03:19:04Z 2013-02-22T03:19:04Z @Martin: Yes. thanks. I get it. http://mathoverflow.net/questions/122589/does-every-metrizable-space-have-point-countable-base Comment by Paul Paul 2013-02-22T02:13:39Z 2013-02-22T02:13:39Z Sorry. What is your meaing? Could you express clearly? http://mathoverflow.net/questions/122528/a-question-on-metrizable-space/122540#122540 Comment by Paul Paul 2013-02-22T00:23:14Z 2013-02-22T00:23:14Z The mathod very good!