Comparison of topologies

In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other. The set of all possible topologies on a given set forms a partially ordered set. This order relation can be used to compare the different topologies.

1 Definition
2 Examples
3 Properties
4 Lattice of topologies
5 Notes
6 See also
7 References

Definition

Let τ₁ and τ₂ be two topologies on a set X such that τ₁ is contained in τ₂:

$\tau_1 \subseteq \tau_2$ .

That is, every element of τ₁ is also an element of τ₂. Then the topology τ₁ is said to be a coarser (weaker or smaller) topology than τ₂, and τ₂ is said to be a finer (stronger or larger) topology than τ₁. ^{[nb 1]} If additionally

$\tau_1 \neq \tau_2$

we say τ₁ is strictly coarser than τ₂ and τ₂ is strictly finer than τ₁.^[1]

The binary relation ⊆ defines a partial ordering relation on the set of all possible topologies on X.

Examples

The finest topology on X is the discrete topology. The coarsest topology on X is the trivial topology.

In function spaces and spaces of measures there are often a number of possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships.

All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.

Properties

Let τ₁ and τ₂ be two topologies on a set X. Then the following statements are equivalent:

τ₁ ⊆ τ₂
the identity map id_X : (X, τ₂) → (X, τ₁) is a continuous map.
the identity map id_X : (X, τ₁) → (X, τ₂) is an open map (or, equivalently, a closed map)

Two immediate corollaries of this statement are

A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer.
An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser.

One can also compare topologies using neighborhood bases. Let τ₁ and τ₂ be two topologies on a set X and let B_i(x) be a local base for the topology τ_i at x ∈ X for i = 1,2. Then τ₁ ⊆ τ₂ if and only if for all x ∈ X, each open set U₁ in B₁(x) contains some open set U₂ in B₂(x). Intuitively, this makes sense: a finer topology should have smaller neighborhoods.

Lattice of topologies

The set of all topologies on a set X together with the partial ordering relation ⊆ forms a complete lattice. That is, any collection of topologies on X have a meet (or infimum) and a join (or supremum). The meet of a collection of topologies is the intersection of those topologies. The join, however, is not generally the union of those topologies (the union of two topologies need not be a topology) but rather the topology generated by the union.

Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element. In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.

Notes

^ There are some authors, especially analysts, who use the terms weak and strong with opposite meaning (Munkres, p. 78).

References

^ Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall. pp. 77–78. ISBN 0-13-181629-2.