2017-11-08 11:54:44 8 Comments

As we know, every regular weakly Lindelof space is DCCC. Here DCCC denotes discrete countable chain condition, a space $X$ has discrete countable chain condition if every discrete family of non-empty open sets of $X$ is countable.

A space $X$ is said to be weakly Lindelof if every open cover $\mathcal U$ of $X$ contains a countable subfamily $\mathcal V \subset \mathcal U$ such that $\bigcup \mathcal V$ is dense in $X$.

Is there a Hausdorff weakly Lindelof space which is not DCCC?

### Related Questions

#### Sponsored Content

#### 0 Answered Questions

### Cellular-Lindelof: a common generalization of the Lindelof property and the CCC

**2017-11-15 10:12:34****Santi Spadaro****176**View**8**Score**0**Answer- Tags: set-theory gn.general-topology

#### 0 Answered Questions

### Is there a normal separable sequential $\aleph$-space with uncountable extent?

**2019-01-29 15:56:16****Taras Banakh****62**View**2**Score**0**Answer- Tags: set-theory gn.general-topology

#### 1 Answered Questions

### [SOLVED] When every open cover admits a $\sigma$-disjoint subcover?

**2018-07-14 08:41:46****MasleniZZa****129**View**1**Score**1**Answer- Tags: gn.general-topology

#### 0 Answered Questions

### Is every weakly Lindelof Banach space a $D$-space?

**2017-12-16 16:29:45****Santi Spadaro****59**View**3**Score**0**Answer- Tags: fa.functional-analysis gn.general-topology

#### 1 Answered Questions

### [SOLVED] Is there a metacompact, normal, CCC space which is not Lindelof

**2017-11-14 02:45:20****Paul****92**View**3**Score**1**Answer- Tags: gn.general-topology counterexamples

#### 1 Answered Questions

### [SOLVED] Is there a calibre $\aleph_1$ Moore space which is not separable

**2017-11-11 08:31:36****Paul****72**View**3**Score**1**Answer- Tags: gn.general-topology counterexamples

#### 0 Answered Questions

### Looking for a weakly Lindel\"of Tychonoff Moore non-ccc space

**2017-05-11 01:09:52****Paul****52**View**2**Score**0**Answer- Tags: gn.general-topology counterexamples

#### 1 Answered Questions

### [SOLVED] On the cardinality of perfect spaces with the countable chain condition

**2013-05-13 21:43:43****Santi Spadaro****478**View**7**Score**1**Answer- Tags: gn.general-topology

#### 2 Answered Questions

### [SOLVED] closed subset of weakly lindelof

**2012-04-21 18:05:09****Douglas Somerset****254**View**3**Score**2**Answer- Tags: gn.general-topology

#### 1 Answered Questions

### [SOLVED] CCC +ă€€collectionwise normality =>ă€€paracompact?

**2011-10-24 13:01:34****Paul****441**View**3**Score**1**Answer- Tags: gn.general-topology

## 1 comments

## @Taras Banakh 2017-11-08 20:09:34

The answer to this problem is negative because of the following

Theorem.If a topological space $X$ is weakly Lindelof, then each discrete (more generally, locally countable) family of open sets in $X$ is at most countable.Proof.Let $\mathcal U$ be a locally countable family of open subset of $X$. Then each point $x\in X$ has a neighborhood $O_x$ meeting at most countably many sets of the family $\mathcal U$. By the weak Lindelof property of $X$ the open cover $\{O_x:x\in X\}$ contains a countable subfamily $\{O_{x}:x\in C\}$ whose union $\bigcup_{x\in C}O_x$ is dense in $X$ and hence intersects each set $U\in\mathcal U$.Assuming that $\mathcal U$ is uncountable and applying the Pigeonhole Principle, we can find a point $x$ of the countable set $C$ such that $O_x$ intersects uncountably many sets of $\mathcal U$. But this contradicts the choice of $O_x$.