relation $\sim$ with $Q\!:X\rightarrow X/\sim$ denoting the quotient map, $X/\sim$ is totally disconnected: As $Q^{-}([x])$ is connected for each $[x]\in X/\sim$ in a component class of $X$, and as any open or closed subset $A\subseteq X/\sim$ is connected iff $Q^{-}(A)$ is open or closed, it must follow that $A$ can only be a singleton.$\qquad\blacksquare$

The next notion of compactness in topological spaces provides an insight of the role of nonempty adherent sets of filters that lead in a natural fashion to the concept of attractors in the dynamical systems theory that we take up next.

**Definition A3.4.** ***Compactness.*** *A topological space $X$ is* *compact* *iff every open cover of $X$ contains a finite subcover of $X$*. *$\qquad\square$*

This definition of compactness has an useful equivalent contrapositive reformulation: *For any given collection of open sets of $X$ if none of its finite subcollections cover $X$, then the entire collection also cannot cover $X$.* The following theorem is a statement of the fundamental property of compact spaces in terms of adherences of filters in such spaces, the proof of which uses this contrapositive characterization of compactness.

**Theorem A3.1.** *A topological space $X$ is compact iff each class of closed subsets of $X$ with finite intersection property has nonempty intersection.* *$\qquad\square$*

**Proof.** *Necessity.* Let $X$ be a compact space. Let $\mathcal{F}=\{ F_{\alpha}\}_{\alpha\in\mathbb{D}}$ be a collection of closed subsets of $X$ with finite FIP, and let $\mathcal{G}=\{ X-F_{\alpha}\}_{\alpha\in\mathbb{D}}$ be the corresponding open sets of $X$. If $\{ G_{i}\}_{i=1}^{N}$ is a nonempty finite subcollection from $\mathcal{G}$, then $\{ X-G_{i}\}_{i=1}^{N}$ is the corresponding nonempty finite subcollection of $\mathcal{F}$. Hence from the assumed finite intersection property of $\mathcal{F}$, it must be true that $$\begin{array}{ccl}
{\displaystyle X-\bigcup_{i=1}^{N}G_{i}} & = & {\displaystyle \bigcap_{i=1}^{N}(X-G_{i})}\qquad(\textrm{DeMorga