Topology

Elements

Topology strips geometry down to its bare essence — it discards length, angle, and area, keeping only the structure of nearness. Its primitive elements:

• Point — the irreducible atom of a space; location without size.
• Open set — the fundamental form; not defined by distance but by its role in the axioms. Every notion of continuity is expressed through open sets alone.
• Topological space — the composed ideal: a set $X$ together with a collection $\tau$ of subsets declared “open,” satisfying the three axioms.

The key topological forms are properties — shapes of space that survive any bending, stretching, or twisting without tearing:

• Connectedness — the space cannot be split; it is whole.
• Compactness — every open cover has a finite subcover; the space is “finite in spirit.”
• Homeomorphism — the topological equivalence: a continuous bijection with a continuous inverse. Spaces related by homeomorphism are the same topologically.

Classification of surfaces

The system of closed surfaces (up to homeomorphism):

• Sphere $S^2$
• Connected sum of tori $T^2 \# T^2 \# \cdots$
• Connected sum of projective planes (non-orientable)

Axiomatic Structure

A topology on a set $X$ is a collection $\tau$ of subsets (“open sets”) satisfying:

1. $\emptyset \in \tau$ and $X \in \tau$.
2. Any union of sets in $\tau$ is in $\tau$ (closed under arbitrary unions).
3. Any finite intersection of sets in $\tau$ is in $\tau$.

From these three axioms, the entire edifice is derived:

Continuity (the key topological concept) is defined without $\varepsilon$-$\delta$: a function $f : X \to Y$ is continuous if and only if $f^{-1}(V) \in \tau_X$ for every $V \in \tau_Y$ — the preimage of every open set is open.

Compactness: $X$ is compact if every open cover has a finite subcover. The Heine-Cantor theorem proves that a continuous function from a compact space is uniformly continuous.

Connectedness: $X$ is connected if it cannot be written as a disjoint union of two nonempty open sets. The image of a connected space under a continuous map is connected.

Homeomorphism: $f : X \to Y$ is a homeomorphism if $f$ is bijective and both $f$ and $f^{-1}$ are continuous. Homeomorphic spaces share every topological property — they are topologically indistinguishable.

Procedures

Constructing Topological Spaces

New spaces are built from old by systematic procedures:

• Product topology: $X \times Y$ with the coarsest topology making projections continuous.
• Subspace topology: a subset $A \subseteq X$ inherits its topology from $X$ — $U \cap A$ for $U$ open in $X$.
• Quotient topology: identify points via an equivalence relation; $U \subseteq X/{\sim}$ is open iff its preimage is open in $X$.

The Möbius strip, torus, and Klein bottle all arise as quotient spaces — gluing edges of a rectangle according to an identification.

Verifying Topological Properties

To verify compactness: exhibit a finite subcover for an arbitrary open cover, or apply Heine-Borel ($X \subseteq \mathbb{R}^n$ is compact iff closed and bounded). To verify connectedness: attempt to separate $X$ into disjoint open sets and show this fails.

Algebraic Invariants

When two spaces might be homeomorphic, compute algebraic invariants that must be preserved:

• Fundamental group $\pi_1(X, x_0)$: loops at $x_0$ up to homotopy, with concatenation. A simply connected space has $\pi_1 = 0$; the torus has $\pi_1 \cong \mathbb{Z}^2$. Apply van Kampen’s theorem when $X$ decomposes into simpler pieces.
• Homology groups $H_n(X)$: detect $n$-dimensional “holes.” Computed via chain complexes from a simplicial or CW decomposition.

If two spaces have different fundamental groups or homology, they cannot be homeomorphic.

Topology as a System

A topological space is a system of open sets closed under the operations prescribed by the axioms. The system’s state is the topology $\tau$; continuous functions are the structure-preserving flows between systems.

• Homeomorphism is the equilibrium relation: two spaces in homeomorphic correspondence are topologically the same. The system has no incentive to distinguish them.
• Homotopy introduces a coarser equilibrium: spaces are homotopy equivalent if one can be continuously deformed into the other. This is a weaker relation — the circle is homotopy equivalent to the annulus but not homeomorphic to it.
• Classification is the system’s fixed taxonomy: the classification theorem for compact surfaces organizes all compact 2-manifolds into equivalence classes by Euler characteristic and orientability. Each class is a stable state of the classification system.

The feedback loop: a topological invariant (fundamental group, homology) takes a space as input and returns an algebraic object. If two spaces yield different invariants, they are distinguishable. The invariants are the system’s observable outputs.