General Topology: Basic Concepts

General Topology: What Does "Continuity" Mean in Full Generality?

Why Is General Topology Needed?

Analysis on $\mathbb{R}$ or $\mathbb{R}^n$ uses a metric (distance) to define continuity, open sets, convergence. But many important spaces do not have a "reasonable" metric: the space of all continuous functions on $[0,1]$ with pointwise topology, the space of measures, spaces in functional analysis.

Topology gives the minimal structure for defining continuity: it suffices to know which sets are "open" — without the concept of distance.

Topological Space

Definition: A pair $(X, \tau)$, where $X$ is a set, $\tau \subset 2^X$ is a family of "open" sets satisfying: $\emptyset, X \in \tau$; the union of any family from $\tau$ is in $\tau$; the intersection of any finite family from $\tau$ is in $\tau$.

Closed set: The complement of an open set.

Examples of topologies on $X$:

Metric topology: $U$ is open $\leftrightarrow$ for all $x \in U$ there exists $B(x, \varepsilon) \subset U$. The standard topology of $\mathbb{R}^n$.

Discrete: $\tau = 2^X$ (all sets are open). Maximally fine.

Trivial: $\tau = {\emptyset, X}$. Maximally coarse.

Zariski: $U$ is open $\leftrightarrow$ $X \setminus U$ is finite (or all $X$). Used in algebraic geometry.

Separation Axioms

How "pointwise" can topological spaces be distinguished?

$T_1$: For any $x \neq y$: there exists $U$ containing $x$ and $y \notin U$ (and symmetrically). Equivalently: ${x}$ is closed for all $x$.

$T_2$ (Hausdorff): For any $x \neq y$: there exist disjoint $U$ containing $x$ and $V$ containing $y$. In a Hausdorff space limits are unique. All metric spaces are $T_2$. Manifolds by definition are Hausdorff.

$T_3$ (regular): A point and a closed set are separated by open sets. $T_3 + T_1 =$ regular Hausdorff.

$T_4$ (normal): Two disjoint closed sets are separated by open sets. Metric spaces are $T_4$. Urysohn's theorem: $T_4 \leftrightarrow$ the existence of continuous separating functions.

Continuity and Homeomorphism

Continuity: $f: X \to Y$ is continuous if the preimage of any open set is open: for all $V \in \tau_Y$, $f^{-1}(V) \in \tau_X$.

This generalizes the $\varepsilon$-$\delta$ definition: for $V = (f(x)-\varepsilon, f(x)+\varepsilon)$ — exactly $\varepsilon$-$\delta$ on $\mathbb{R}$.

Homeomorphism: A bijective continuous function with continuous inverse. Homeomorphic spaces are "the same" from the point of view of topology.

Topological invariant: A property preserved under homeomorphism. The number of connected components, compactness, Hausdorffness — invariants. Examples: sphere $S^2$ and torus $T^2$ are not homeomorphic (different fundamental groups). The segment $[0,1]$ and the circle $S^1$ are not homeomorphic (boundary points).

Product and Quotient Spaces

Product $X \times Y$: $\tau_{X\times Y}$ is generated by rectangles $U \times V$ ($U \in \tau_X$, $V \in \tau_Y$). Universal property: $f: Z \to X \times Y$ is continuous $\leftrightarrow$ $\pi_X \circ f$ and $\pi_Y \circ f$ are continuous.

Tikhonov's Theorem: Arbitrary product of compact spaces is compact (assuming the axiom of choice).

Quotient space $X/\sim$ (identification by equivalence): open sets — preimages of open sets in $X$. Examples: $[0,1]/{0 \sim 1} \cong S^1$ (glue the ends of the segment $\to$ circle). $D^2/\partial D^2 \cong S^2$ (collapse the disk by gluing all the boundary into a point). $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ (the torus as a quotient of the plane).

Numerical Examples and Real Applications

Configuration spaces in robotics: Position of a pendulum — $S^1$. Position of a double pendulum — $T^2$. Position of a rotation link — $SO(3)$. The robot's configuration space is a direct product. Motion planning uses topology of these spaces: Is there a continuous path from the start configuration to the target?

Phase transitions: The space of order parameters has a defined topology. For a ferromagnet — $S^2$ (direction of magnetization). Topological defects (domain walls, vortices, monopoles) are classified by the fundamental group $\pi_1(M)$, $\pi_2(M)$ of the parameter space.

Topology in Data Analysis

Topological Data Analysis (TDA) transfers ideas of general topology to machine learning tasks. Persistent homology tracks topological features of a point cloud (connected components $\beta_0$, "loops" $\beta_1$, "voids" $\beta_2$) as the neighborhood radius changes. Features that "survive" over a wide range of scales are considered significant structural traits of the data, not noise. Used in biology (protein molecule shape), neuroscience (topology of neuronal activations), materials science (porosity analysis). TDA detects topological signatures of data without assumptions about smoothness or dimensionality.

Topological Spaces in Economics and Game Theory

General topology appears unexpectedly in purely economic results. Debreu's theorem on the existence of competitive equilibrium relies on Kakutani's fixed point theorem — a consequence of the compactness of the simplex of strategies and continuity of demand functions. It is precisely the topological property of compactness that ensures the existence of an equilibrium price. Brouwer's fixed point theorem — a continuous function from a compact convex set to itself has a fixed point — underlies the proof of Nash equilibrium existence in mixed strategies, equilibrium theorems in general equilibrium theory (Arrow–Debreu), as well as a number of results in the theory of differential equations. In dynamical systems theory, attracting invariant sets (attractors) possess nontrivial topological structure: strange attractors of chaotic systems are fractals with a non-integer Hausdorff dimension. Topological classification of attractors determines the type of dynamics: equilibrium — point, limit cycle — circle, torus corresponds to quasiperiodic motion, and chaotic attractor — an object with nontrivial topology.

Topological Spaces in Data Analysis and Network Science

Topological concepts of continuity and compactness find direct applications in modern mathematics of data. Alexander–Subbasis theorem gives a compactness criterion via subbasis covers — the foundation of the ultrafilter method, used in nonstandard analysis and theoretical computer science. In computer sciences, the Zariski space models the semantics of programming languages through closed sets of satisfaction points for predicates. The topology of the Internet and social networks is analyzed through metric properties and compactness: the "small world" of Watts–Strogatz describes a graph with small diameter and high clustering coefficient — a property close to local compactness. In theoretical computer science, Scott topological spaces are used to describe the semantics of recursive definitions and domain equations. In bioinformatics, topological methods are applied for classification of molecule and protein shapes: the metric space of protein conformations with RMSD distance is used for clustering structural ensembles via molecular dynamics.

Exercise: (a) For $T = [0,1]/{0 \sim 1} \cong S^1$: explicitly describe a continuous bijection $[0,1)/\sim \to S^1$. (b) The space $\mathbb{R}$ with Zariski topology: is it compact? Is it Hausdorff? (c) Is the mapping $f(t) = e^{2\pi i t}: \mathbb{R} \to S^1$ continuous? Is it a homeomorphism? Why?