Compactness is a central, but complex notion in analysis. I'm finding more concise definition of compact spaces. The one I easily found is the definition by 'finite intersection property', but it doesn't quite make the situation simpler.
As it is seen in nlab, there is (seemingly) very concise definition of compact space, by compact object in category theory. Directly unwinding the definition, it looks the most complicated: $X$ is compact if and only if the functor $$\mathrm{hom}(X,-)$$ preserves filtered colimits in the category of open sets $\mathrm{Op}(X)$.
If we look closer, since the category $\mathrm{Op}(X)$ is merely a poset, we can translate it as:
For any filtered family of open sets
$\left\{ X_i\right \}$, $X\subseteq \bigcup X_i$ if and only if $X \subseteq X_i$ for some $i$.
Which is in turn, a definition of compact object in a poset (if your familiar with it). Again, the notion of 'filteredness' is quite involved. So my aim is to lessen the condition.
The definition of 'compact object' simply means that there are no nets of proper open subsets of $X$ that unions to $X$. I want to make the net merely to be a chain.
Here is my reasoning: If a space $X$ has a chain of proper open subsets that unions to $X$, it's clearly not compact. Conversely, if $X$ is not compact, we have an open cover $\left\{U_i\right\} $ of $X$ not admitting a finite open subcover. (With Axiom of Choice) we can well-order $\left\{U_i\right\}$ and we'll get a sequence of open subsets $\left\{U_\alpha\right\}_{\alpha<\beta}$.
First, just naively construct the increasing sequence $V_\alpha=\bigcup_{\gamma<\alpha} U_\gamma$, which unions to $X$. Suppose the sequence stabilizes from the $\beta_0$th step ($\beta_0$ can be chosen to be the least one). Then $\beta_0$ should be infinite. If $\beta_0$ is limit, $\left\{V_\alpha\right\}_{\alpha<\beta_0}$ is the desired sequence. Otherwise, $\beta_0=\beta_0'+1$, and $V_{\beta_0'}$ should be $X$ as $V_\alpha$ is increasing. We now move $U_{\beta_0'}$ to the very first of the sequence $\left\{U_\alpha\right\}$ and still we get a well-ordered sequence.
Now, retrying all the process, we get a $\beta_1$, which is the smallest ordinal $\beta$ among $\bigcup_{\alpha<\beta} U_\alpha=X$. $\beta_1$ is strictly smaller than $\beta_0$, since $\bigcup_{\alpha<\beta_0'} U_\alpha = X$ (with the new sequence $\{U_\alpha\}$. Repeating this process, we can do an infinite descent of ordinals, so it should end with a limit ordinal, getting the desired chain.
Is my reasoning OK? and can I find such a definition in a Literature?