There is a large history in defining precisely what a topology is, but the modern and standard definition is one that relies upon open sets.
A topology on a set X is a collection \mathcal{T} of subsets of X called open sets, which have the following properties:
(O1) \colon The empty set \varnothing and the set itself X are in \mathcal{T}.
(O2) \colon Any union of elements in \mathcal{T} is in \mathcal{T}.
(O3) \colon Any finite intersection of elements in \mathcal{T} is in \mathcal{T}.
A topological space is an ordered pair (X,\mathcal{T}) such that X is a set and \mathcal{T} is a topology on X.
Note that this definition gives no insight into what an open set is, just properties we expect open sets to have. Given the same set X, we can construct many different topologies by choosing different sets to be open.
Let (X,\mathcal{T}) be a topological space, a subset F of X is called closed if F^C is open, or F^C \in \mathcal{T}.
By applying De Morgan laws to our open set properties, we can easily show the following lemma.
Let (X,\mathcal{T}) be a topological space. Then the following are true:
(C1) \colon The empty set \varnothing and the set itself X are closed.
(C2) \colon Any finite union of closed sets is closed.
(C3) \colon Any intersection of closed sets is closed.
Comparing Topologies
We have already seen how the same set X can be given many separate topologies by choosing different sets to be open. The following definition gives us a way to compare the ‘size’ of different topologies.
Let \mathcal{T} and \mathcal{T}^\prime be topologies on a set X.
If \mathcal{T} \subseteq \mathcal{T}^\prime, we say that \mathcal{T}^\prime is finer than \mathcal{T}.
If \mathcal{T^\prime} \subseteq \mathcal{T}, we say that \mathcal{T} is coarser than \mathcal{T}^\prime.
If \mathcal{T} \subset \mathcal{T}^\prime, we say that \mathcal{T}^\prime is strictly finer than \mathcal{T}.
If \mathcal{T^\prime} \subset \mathcal{T}, we say that \mathcal{T} is strictly coarser than \mathcal{T}^\prime.
In his book, “Topology”, James Munkres gives the following quote as a helpful way to think about the nomenclature.
This terminology is suggested by thinking of a topological space as being something like a truckload full of gravel—the pebbles and all unions of collections of pebbles being the open sets. If now we smash the pebbles into smaller ones, the collection of open sets has been enlarged, and the topology, like the gravel, is said to have been made finer by the operation.