Comparison of topologies

An order relation on the set of all topologies on one and the same set $ X $. A topology $ {\mathcal T} _ {1} $ majorizes a topology $ {\mathcal T} _ {2} $( or $ {\mathcal T} _ {1} $ is not weaker than $ {\mathcal T} _ {2} $), if the identity mapping $ X _ {1} \rightarrow X _ {2} $, where $ X _ {i} $ is the set $ X $ with the topology $ {\mathcal T} _ {i} $, $ i = 1, 2 $, is continuous. Moreover, if $ {\mathcal T} _ {1} \neq {\mathcal T} _ {2} $, then $ {\mathcal T} _ {1} $ is stronger than $ {\mathcal T} _ {2} $( or $ {\mathcal T} _ {2} $ is weaker than $ {\mathcal T} _ {1} $).

The following statements are equivalent:

1) $ {\mathcal T} _ {1} $ majorizes $ {\mathcal T} _ {2} $.

2) For any $ x \in X $, every neighbourhood of $ x $ in the topology $ {\mathcal T} _ {2} $ is a neighbourhood of $ x $ in the topology $ {\mathcal T} _ {1} $.

3) For any $ A \subset X $, the closure of $ A $ in $ {\mathcal T} _ {2} $ contains the closure of $ A $ in $ {\mathcal T} _ {1} $.

4) Every set from $ X $, closed in $ {\mathcal T} _ {2} $, is also closed in $ {\mathcal T} _ {1} $.

5) Every set that is open in $ {\mathcal T} _ {2} $ is open in $ {\mathcal T} _ {1} $.

In the ordered set of topologies on $ X $, the discrete topology is the strongest, while the topology whose only closed sets are $ \emptyset $ and $ X $ is the weakest. Figuratively speaking, the stronger the topology, the more open sets, closed sets and neighbourhoods there are in $ X $; the stronger the topology, the smaller the closure of a set (and the larger its interior) and the smaller the number of everywhere-dense sets.