Topology

Topological Spaces

A coffee cup and a donut are the same object. To a topologist. Both have exactly one hole, and one deforms into the other without tearing. A sphere and a cube are also the same. But a sphere and a torus are different. This is not wordplay. This exact distinction is used in topological data analysis (TDA) to find structure in high-dimensional genomic and neural data.

  • **Topological Data Analysis (TDA):** persistent homology finds 'holes' in point clouds in hundreds of dimensions - applied in genomics (cancer clusters), neuroscience (shape of neural activation patterns), and materials science
  • **Mapper algorithm:** UMAP and t-SNE compress dimensions linearly; Mapper (Carlsson 2009) builds a topological graph of data - used to analyze brain activity patterns and predict cancer recurrence
  • **Robotics:** the robot configuration space is a topological space; motion planning = finding a path avoiding obstacles in this space

Open Sets

In analysis, the open interval (0, 1) is a set where every point is surrounded by a 'buffer zone' lying entirely inside the set. Topology generalizes this intuition: an **open set** is a set declared to be open, provided the collection of such sets satisfies certain axioms.

**Three axioms of open sets:** 1. The empty set and the whole space X are open. 2. The union of any number of open sets is open. 3. The intersection of finitely many open sets is open. Why only finite? Because an infinite intersection of open sets may fail to be open.

A critical point: the notion of 'open' is not absolute - it is defined by the chosen topology. The same set can be open in one topology and not open in another. This freedom of choice is what makes topology so powerful.

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A closed set is the complement of an open set. In the standard topology on R, the interval [0,1] is closed because R \ [0,1] = (-inf, 0) union (1, inf) is open. But a set can be simultaneously open and closed (clopen), or neither.

Why does the axiom require finite intersections rather than arbitrary ones?

Topology on a Set

A **topology** tau on a set X is a collection of subsets of X satisfying the three open-set axioms. The pair (X, tau) is called a **topological space**. Different tau on the same X yield different spaces with different properties.

**Three canonical examples:** 1. **Trivial** (indiscrete) topology: tau = {empty, X} - only two open sets. 2. **Discrete** topology: tau = P(X) - all subsets are open. 3. **Standard** topology on R: open sets are unions of open intervals.

The trivial topology is the 'coarsest': almost nothing can be distinguished in it. The discrete topology is the 'finest': every point is separated from the rest. The finer the topology, the more open sets there are, and the more continuous functions exist FROM that space.

TopologyOpen setsProperties
Trivial{empty, X}Compact, connected; not Hausdorff (if |X|>1)
DiscreteAll subsetsHausdorff; compact only for finite X
Standard on RUnions of (a,b)Hausdorff, connected, not compact
ZariskiComplements of finite setsNot Hausdorff; used in algebraic geometry

Felix Hausdorff and the Birth of Topology

In 1914, Felix Hausdorff published 'Grundzüge der Mengenlehre', in which he systematically defined topological spaces via neighborhood axioms for the first time. His definition (Hausdorff spaces) turned out to be too restrictive, and the modern version based on open sets became the standard.

On the set X = {a, b}, the collection tau = {empty, {a}, {b}, {a,b}} is which topology?

Basis of a Topology

Describing a topology by listing all open sets is inconvenient - there may be infinitely many. A **basis** is a compact way to specify a topology: a collection of 'basic' open sets whose unions can generate every open set.

**Basis B** for a topology tau: 1. For each point x in X there exists B in B such that x in B. 2. If x in B1 inter B2, there exists B3 in B such that x in B3 subset B1 inter B2. The topology generated by B: a set is open if it is a union of elements of B.

Different bases can generate the same topology. Open intervals (a, b) and open intervals with rational endpoints (p, q), p, q in Q, generate the same standard topology on R. But the second basis is countable!

SpaceBasisSize of basis
R, standardOpen intervals (a,b)Uncountable
R, standardIntervals with rational endpoints (p,q)Countable
R^n, standardOpen balls B(x, r)Uncountable
Discrete on XSingletons {x}|X|
Zariski on RR \ {finite set}Countable

Why does {(a,b) : a < b, a,b in Q} also form a basis for the standard topology on R?

Subspace Topology

If (X, tau) is a topological space and A subset X, how do we define a topology on A? The **induced (subspace) topology**: a set U subset A is open in A if and only if U = A inter V for some open V in X.

**tau_A = {A inter V : V in tau}**. This is always a topology on A (verified by direct substitution into the axioms). The open sets of A are the 'traces' of open sets of X on A.

An important consequence: a set can be open in a subspace but not in the ambient space. [0, 0.5) is open in [0,1] (= [0,1] inter (-1, 0.5)), but not open in R. Properties depend on context - which topology we are working in.

Subspace A of RInduced topologyNote
(0,1)Coincides with standard on (0,1)Open subset of R
[0,1][0,a) and (b,1] are also openClosed subset of R
Z (integers)DiscreteEvery point is isolated
Q (rationals)Totally disconnectedIrrationals lie between any two rationals

The subspace topology is the natural way to speak about subsets as spaces in their own right. The sphere S^2 is a subspace of R^3. The circle S^1 is a subspace of R^2. Their topological properties are inherited from the ambient space via the induced topology.

An open set = a non-closed set

A set can be simultaneously open AND closed (clopen). Examples: the empty set and X itself are always clopen in any topology. In the discrete topology, EVERY subset is clopen. A set can also be neither open nor closed: [0, 1) in R

'Open' and 'closed' are not opposites - they are two independent properties. Closed = complement is open. A set is clopen if both it and its complement are open. In TDA this distinction matters: the persistent homology algorithm works with filtrations of open and closed covers - and clopen sets appear as connected components at the coarsest scale

The set [0, 0.5) in the subspace [0, 1] of R is:

Key Ideas

  • **Open set** defined by axioms: $\emptyset$ and $X$ are open, unions of any number of open sets are open, finite intersections are open - finite because $\cap_{n=1}^\infty (-1/n, 1/n) = \{0\}$ is no longer open
  • **Topology** $\tau$ on $X$ - a collection of open subsets; the choice of topology determines what 'nearby' and 'continuous' mean
  • **Basis** generates the topology: the standard topology on $\mathbb{R}$ is generated by open intervals with rational endpoints - a countable basis
  • **Subspace topology** $\tau_A = \{A \cap V : V \in \tau\}$ - openness depends on context; $[0, 0.5)$ is open in $[0,1]$ but not in $\mathbb{R}$
  • **TDA link:** persistent homology filters the topological space by scale and records which 'holes' are born and die - this is what detects clusters and loops in cancer and brain data

Related Topics

Open sets are the foundation of all of topology:

  • Continuity and Homeomorphism — Continuity is defined via open sets; homeomorphism is topological equivalence
  • Connectedness and Compactness — Defined via open sets; key invariants of spaces

Вопросы для размышления

  • Why is every function f: Y → X continuous in the trivial topology? What does this say about the 'distinguishing power' of the trivial topology?
  • If axiom 3 allowed infinite intersections, which topologies would remain? (Hint: only the discrete one.)
  • The set Z (integers) with the topology induced from R is discrete. Why?

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Topological Spaces