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 X 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.

Definition

Given a set X we define a relation ⊆ between two topologies τ₁ and τ₂ on X

<math>\tau_1 \subseteq \tau_2<math>

if τ₂ contains all the open sets of τ₁. This relation defines a partial ordering relation on the set of all possible topologies on X.

We say that τ₂ is finer (stronger or larger) topology than τ₁ and τ₁ is coarser (weaker or smaller) topology than τ₂.

If additionally

<math>\tau_1 \neq \tau_2<math>

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

Alternatively we say τ₁ is coarser than τ₂ if the identity mapping

<math>i: (X,\tau_2) \to (X,\tau_1)<math>

is continuous.

Be aware that there are some authors, esp. analysts, who use the terms weak and strong with opposite meaning.

Examples

The finest topology on X is the discrete topology. The coarsest topology on X is the trivial topology. Any two topologies on X have a meet and join, in the sense of lattice theory; the meet is the intersection, but the join is not in general the union.

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

• Given a continuous functions f between two topological space X and Y then f stays continuous if the topology on Y becomes coarser or the topology on X finer.

• If τ₁ is coarser than τ₂ then every open (closed) set in τ₂ is open (closed) in τ₁

See also

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