Topology: Open Sets, Continuity, Compactness
The abstraction of "closeness" without distances — and what continuous maps preserve.
A topology on a set $X$ is a collection $\tau$ of subsets (the open sets) with:
- $\emptyset, X \in \tau$.
- Arbitrary unions of open sets are open.
- Finite intersections of open sets are open.
A map $f: X \to Y$ between topological spaces is continuous if preimages of open sets are open.
Key invariants under homeomorphism (continuous bijection with continuous inverse):
- Compactness: every open cover has a finite subcover. Equivalent to "closed and bounded" in $\mathbb R^n$ (Heine–Borel).
- Connectedness: cannot be split into two disjoint nonempty open sets.
- Hausdorff: distinct points have disjoint open neighborhoods.
Continuous images of compact sets are compact; continuous images of connected sets are connected.