Editing Comparison of topologies (section)

== Definition ==
A topology on a set may be defined as the collection of [[subset]]s which are considered to be "open". (An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the [[Complement (set theory)|complement]] of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.) 

For definiteness the reader should think of a topology as the family of '''open sets''' of a topological space, since that is the standard meaning of the word "topology".

Let ''τ''<sub>1</sub> and ''τ''<sub>2</sub> be two topologies on a set ''X'' such that ''τ''<sub>1</sub> is contained in ''τ''<sub>2</sub>:
:<math>\tau_1 \subseteq \tau_2</math>.
That is, every element of ''τ''<sub>1</sub> is also an element of ''τ''<sub>2</sub>. Then the topology ''τ''<sub>1</sub> is said to be a '''coarser''' ('''weaker''' or '''smaller''') '''topology''' than ''τ''<sub>2</sub>, and  ''τ''<sub>2</sub> is said to be a '''finer''' ('''stronger''' or '''larger''') '''topology''' than ''τ''<sub>1</sub>.
<ref group="nb">There are some authors, especially [[mathematical analysis|analyst]]s, who use the terms ''weak'' and ''strong'' with opposite meaning. {{harv|Munkres|2000|p=78}}</ref>

If additionally 
:<math>\tau_1 \neq \tau_2</math>
we say ''τ''<sub>1</sub> is '''strictly coarser''' than  ''τ''<sub>2</sub> and ''τ''<sub>2</sub> is '''strictly finer''' than ''τ''<sub>1</sub>.{{sfn|Munkres|2000|pp=77-78}}

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