Comparison of topologies

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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.

Contents 1 Definition 2 Examples 3 Properties 4 Lattice of topologies 5 See also

[edit] Definition

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

.

That is, every set open under τ₁ is also open under 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. If additionally

we say τ₁ is strictly coarser than τ₂ and τ₂ is strictly finer than τ₁.

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

N.B. There are some authors, especially analysts, who use the terms weak and strong with opposite meaning.

[edit] 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.

[edit] 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.

[edit] 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 is 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.

[edit] See also

• Initial topology, the coarsest topology on a set to make a family of mappings from that set continuous
• Final topology, the finest topology on a set to make a family of mappings into that set continuous