An example of an uncountable, non-compact, non-metrizable, d-compact space 
A space is said to be d-compact if every cover by open $F_\sigma$ sets has a finite subcover.

I am looking for an example of an uncountable, non-compact, non-metrizable space which is d-compact.
I have looked into Fortissimo space, which is defined as a space $X$ with a point $p$ in it, and in which open sets are the ones whose complement is either countable or contains the point $p$.
Taking $X=R$ and $p=0$, I get that closed sets are either countable ones or the ones with $0$ in it. Also $F_\sigma$ sets are either countable sets or uncountable sets with $0$ in it. Hence open $F_\sigma $ sets are uncountable sets with $0$ in it such that their complement is countable.
But to check whether it is d-compact or not, is what I am stuck at? If it is not, can someone suggest some other topological space with above mentioned properties? Thanks!
 A: The uncountable Fortissimo is not d-compact, and the proof relies on a characterization of the Fσ-sets in the space. As you have noticed, $F \subseteq X$ is closed iff either (1) $F$ is countable, or (2) $p \in F$.
Suppose that $A = \bigcup_{n \in \mathbb{N}} F_n \subseteq X$ is Fσ, where each $F_n$ is closed. If $p \in F_n$ for some $n$, then clearly $p \in A$, and so $A$ is closed. If $p \notin F_n$ for each $n$, then each $F_n$ is countable, and so $A = \bigcup_{n \in \mathbb{N}} F_n$ is also countable, and hence closed.
Therefore the Fσ-sets in $X$ coincide with the closed subsets of $X$. It immediately follows that the open Fσ-sets in $X$ coincide with the clopen subsets of $X$.
That is $A \subseteq X$ is an open Fσ-set iff either (1) $p \notin A$ and $A$ is countable, or (2) $p \in A$ and $X \setminus A$ is countable.
Pick a countable $B = \{ x_n : n \in \mathbb{N} \} \subseteq X \setminus \{ p \}$. Then $X \setminus B$ is a clopen subset of $X$, and for each $n$ the singleton $\{ x_n \}$ is also clopen. Clearly $( X \setminus B ) \cup \bigcup_{n \in \mathbb{N}} \{ x_n \} = X$, however there is no finite subcover (in fact, there is no proper subcover).

An example of an uncountable d-compact space which is not compact and not metrizable is the particular point topology on an uncountable set. Fix an uncountable set $X$, and fix some $p \in X$. The open subsets of $X$ are $\emptyset$, and all subsets of $X$ which contain $p$.


*

*This space is not compact because fixing some countable subset $A = \{ x_n : n \in \mathbb{N} \} \subseteq X \setminus \{ p \}$, we have that $\{ X \setminus A \} \cup \{ \{ p , x_n \} : n \in \mathbb{N} \}$ is an open cover with no proper subcover (and so no finite subcover).

*This space is not metrizable because it is not even T1 (there is no open subset which contains $x \in X \setminus \{ p \}$ but not $p$).

*This space is d-compact because the only open Fσ-sets are $\emptyset$ and $X$. (The only closed subset of $X$ which contains $p$ is $X$, and so the only Fσ-subset of $X$ which contains $p$ is $X$.)
