# Comparison of topologies

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for **comparison of the topologies**.

## Definition

Let τ_{1} and τ_{2} be two topologies on a set *X* such that τ_{1} is contained in τ_{2}:

That is, every element of τ_{1} is also an element of τ_{2}. Then the topology τ_{1} is said to be a **coarser** (**weaker** or **smaller**) **topology** than τ_{2}, and τ_{2} is said to be a **finer** (**stronger** or **larger**) **topology** than τ_{1}.
^{[nb 1]}
If additionally

we say τ_{1} is **strictly coarser** than τ_{2} and τ_{2} is **strictly finer** than τ_{1}.^{[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; this topology makes all subsets open. The coarsest topology on *X* is the trivial topology; this topology only admits the null set
and the whole space as open sets.

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 τ_{1} and τ_{2} be two topologies on a set *X*. Then the following statements are equivalent:

- τ
_{1}⊆ τ_{2} - the identity map id
_{X}: (*X*, τ_{2}) → (*X*, τ_{1}) is a continuous map. - the identity map id
_{X}: (*X*, τ_{1}) → (*X*, τ_{2}) 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 τ_{1} and τ_{2} 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 τ_{1} ⊆ τ_{2} 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 also closed under arbitrary intersections. 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

## 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

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}