# Concerning a strongly nowhere dense subset

A pair $(X,\tau )$ is called a generalized topological space if $\tau$ is collection of subsets of $X$ so that $\emptyset \in \tau$ and $\tau$ is closed under arbitrary unions. A subset $A$ of GTS $(X,\tau )$ is strongly nowhere dense if for any nonempty open set $U\in \tau$, there exists nonempty open set $V \subset U$ such that $V\cap A=\emptyset$.

I need an example of a generalized topological space $X$ (which is not topological space) contains a strongly nowhere dense subset?

• You want your strongly nowhere dense open subset to be non-empty, right? – Dominic van der Zypen Oct 23 '17 at 12:38

I add another answer in light of your comment to the answer above. Here I construct a generalized topological space (which is not a topological space) along with a non-empty strongly nowhere dense subset.

Let $X = \{0,1,2,3\}$ and let $$\tau = \big\{\{0,1\}, \{1,2\}, \{0,1,2\}, X\big\}.$$

Note that $X$ is the only member of $\tau$ containing $3$. It is easy to see that $(X,\tau)$ is a generalized topological space, but it is not a topological space, because $\{0,1\}, \{1,2\}\in \tau$, but $\{1\} = \{0,1\}\cap \{1,2\} \notin \tau$.

Let $A = \{3\}$. It is easy to see that $A$ is strongly nowhere dense.

• That's great. Thank you for your answer. I sincerely apologize that I had a mistake in my question. Thank you for taking the trouble to help me. – MHenry Oct 23 '17 at 15:19
• You are welcome! Can you press +1 next to my answer if you found it useful? – Dominic van der Zypen Oct 23 '17 at 17:45

Let $(X,\tau)$ be any generalized topological space. If $A\subseteq X$ is open and $A\neq \emptyset$, it cannot be be strongly nowhere dense: Let $U=A$, then for every nonempty open set $V\subseteq U$ we have $V \cap A = V\cap U = V \neq \emptyset$.

So $A = \emptyset$ is the only strongly nowhere dense subset of $(X,\tau)$.

• I'm sorry. I mean that the wanted strongly nowhere subset in the question must be nonempty. – MHenry Oct 23 '17 at 13:12
• Certainly it is not also open subset. – MHenry Oct 23 '17 at 13:13
• A GTS $(X,\tau )$ is called a weak Baire space if there is no nonempty open set $U\in \tau$ such that can be written as a countable union of strongly nowhere dense subsets. Could you give an example of weak Baire space which is not Baire space? – MHenry Oct 23 '17 at 15:48
• I am away tomorrow but you could ask this as a separate question ... – Dominic van der Zypen Oct 23 '17 at 17:44