Topology: Foundations¶
Job: axiomatic open-set topology — continuity, separation, compactness, connectedness — as the common language of analysis and geometry.
Core¶
Definition. A topology on \(X\) is a family \(\tau\) of subsets (opens) closed under arbitrary unions and finite intersections; \(\emptyset,X\in\tau\). Complements of opens are closed. A map \(f:X\to Y\) is continuous iff \(f^{-1}(V)\) open for all open \(V\subseteq Y\) (equiv. closed-preimage; equiv. \(f(\overline{A})\subseteq\overline{f(A)}\)).
| Notion | Definition |
|---|---|
| Basis / subbasis | generate topology by unions (resp. finite \(\cap\) then unions) |
| Product topology | coarsest making projections continuous (box = all products of opens: finer, rarely used for infinite products) |
| Subspace | opens \(=U\cap Y\), \(U\) open in \(X\) |
| Quotient | \(U\subseteq X/{\sim}\) open iff \(\pi^{-1}(U)\) open |
| Interior / closure | largest open in \(A\) / smallest closed containing \(A\) |
| Dense | \(\overline{A}=X\); separable = countable dense subset |
| 1st / 2nd countable | countable local bases / countable basis |
Separation axioms¶
| Axiom | Meaning |
|---|---|
| \(T_0\) | points distinguished by opens |
| \(T_1\) | singletons closed |
| Hausdorff \(T_2\) | disjoint opens separate points; limits unique |
| Regular \(T_3\) | point and closed set (not containing it) separated by opens |
| Normal \(T_4\) | disjoint closed sets separated by opens |
| Completely regular \(T_{3\frac12}\) | continuous \([0,1]\)-functions separate point/closed set |
Metric \(\Rightarrow\) paracompact Hausdorff + normal. Manifolds: usually require Hausdorff + second-countable.
Urysohn / Tietze (normal spaces). Disjoint closed sets separated by continuous \(f:X\to[0,1]\); continuous functions on closed subsets extend.
Compactness¶
\(X\) compact if every open cover has a finite subcover. Equivalent (Hausdorff): closed subsets of compact are compact; continuous image of compact is compact; closed map \(X\to Y\) with compact fibers has properties used in properness.
| Theorem | Statement |
|---|---|
| Tube lemma | \(X\) compact, \(N\) open nbhd of \(X\times\{y\}\) \(\Rightarrow\) tube \(X\times V\) |
| Tychonoff | arbitrary products of compact spaces are compact (AC) |
| Finite intersection property | compact \(\Leftrightarrow\) families of closed sets with FIP have nonempty \(\cap\) |
Locally compact Hausdorff: every point has a compact nbhd. One-point compactification \(\alpha X=X\cup\{\infty\}\): opens are opens of \(X\) or complements of compact subsets of \(X\). Stone–Čech \(\beta X\) (completely regular): universal compact Hausdorff target for maps from \(X\).
Connectedness¶
\(X\) connected if not union of two disjoint nonempty opens. Continuous image of connected is connected. Path-connected \(\Rightarrow\) connected; converse fails (topologist’s sine curve). Locally path-connected + connected \(\Rightarrow\) path-connected. Components vs path-components vs quasicomponents.
Nets and filters (optional quals)¶
Sequences suffice in first-countable spaces. A net \((x_\lambda)_{\lambda\in\Lambda}\) converges to \(x\) if for every nbhd \(U\ni x\), eventually \(x_\lambda\in U\). Filters / ultrafilters: Tychonoff via ultrafilter compactness; \(\beta X\) as ultrafilters on \(X\) (discrete case).
See it¶
Zariski topology on \(\operatorname{Spec} R\) or \(\mathbb{A}^n_k\): closed sets are zero loci. Not Hausdorff; compact in the sense of quasi-compact (no Hausdorff). Generic points \(\eta\) with \(\overline{\{\eta\}}=X\) for irreducible \(X\) — standard in algebraic geometry, pathological from metric intuition.
Check¶
Prove: Hausdorff + compact subspace \(K\subseteq X\) \(\Rightarrow\) \(K\) closed.
For \(x\notin K\), separate \(x\) from each \(y\in K\) by disjoint opens; finite subcover of the \(K\)-side yields a nbhd of \(x\) missing \(K\).
Counterexample: product of quotient maps need not be quotient.
Standard: \(\mathbb{R}\to\mathbb{R}/\mathbb{Z}\) (or more carefully crafted identification maps); see Munkres. Moral: beware \(\prod\) and quotients interacting.
Pitfalls¶
- Sequential continuity \(\neq\) continuity without first countability.
- Compact \(\not\Rightarrow\) sequentially compact and conversely in general spaces.
- Open map / closed map / quotient map are independent notions.
- CW / manifold hypotheses silently add second-countable + Hausdorff; dropping Hausdorff yields line with two origins — not a manifold in the usual sense.
Refs¶
- Munkres, Topology
- Bourbaki, General Topology
- Willard, General Topology
Next¶
Metric spaces · Real analysis · Algebraic topology · Topology overview · Category theory