I am learning about the almost compact space and nearly compact space. I know that every nearly compact space is almost compact space but the converse is not true in general. So i need an example of almost compact space which is not nearly compact space.
Nearly compact: A space is called nearly compact if each open cover of the space has a finite subfamily the interiors of the closures of whose member cover the space.
Almost compact: A space is called almost compact if each open cover of the space has a finite subfamily the closures of whose member cover the space.
I hope that the followings works. Let $X := \{a, a_n, b_n, c_n, c: n ∈ ω\}$ such that all the points are distinct. Let us consider the topology generated by sets $U_n := \{a_n, b_n, c_n\}$ for $n ∈ ω$, $A := \{a, a_n: n ∈ ω\}$, $C := \{c_n, c: n ∈ ω\}$.
$X$ is almost compact since any open cover contains the sets $A$, $C$ and $\overline{A} ∪ \overline{C} = X$.
$X$ is not nearly compact since the open cover $\{A, U_n, C: n ∈ ω\}$ has no finite subcollection whose regularizations cover $X$ since all the covering sets are regularly open (the interior of the closure is the original set).
In fact the space is semiregular but not compact ($\{b_n: n ∈ ω\}$ is a closed discrete subset), and hence not nearly compact.