Editorially prepared introduction for this redesigned journal. This is new explanatory content, separate from Pablo's original notes.
1. Topologies
Topology studies properties preserved by continuous deformations, beginning with a specification of which sets are open.
Definition. A topology \(\tau\) on \(X\) contains \(\varnothing\) and \(X\), is closed under arbitrary unions, and is closed under finite intersections. The pair \((X,\tau)\) is a topological space.
2. Closed sets
A set \(F\subseteq X\) is closed if its complement \(X\setminus F\) is open. A set can be both open and closed.
3. Continuity
A function \(f:X\to Y\) is continuous if the inverse image \(f^{-1}(U)\) of every open set \(U\subseteq Y\) is open in \(X\).
4. Compactness
Definition. A space is compact if every open cover has a finite subcover.
In Euclidean space, the Heine–Borel theorem characterizes compact sets precisely as the closed and bounded subsets.