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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

Definition

Examples

Properties

Lattice of topologies

See also

Definition

Let τ₁ and τ₂ be two topologies on a set ''X'' such that τ₁ is contained in τ₂:
:$au\_1\; subseteq\; au\_2$.
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
:$au\_1$
eq au_2
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.

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 ''U''₁ in ''B''₁(''x'') contains some open set ''U''₂ in ''B''₂(''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 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.

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