Topology

Once you have understood metric spaces you can read motivation for topology and see why topological spaces "make sense" and extend metric spaces.

Phrases

Let [ilmath](X,\mathcal{J})[/ilmath] and [ilmath](X,\mathcal{K})[/ilmath] be two topologies on [ilmath]X[/ilmath]

Coaser, Smaller, Weaker

Given two topologies [math]\mathcal{J}[/math], [math]\mathcal{K}[/math] on [ilmath]X[/ilmath] we say:
[math]\mathcal{J}[/math] is coaser, smaller or weaker than [math]\mathcal{K}[/math] if [math]\mathcal{J}\subset\mathcal{K}[/math]

Smaller is a good way to remember this as there are 'less things' in the smaller topology.

Finer, Larger, Stronger

Given two topologies [math]\mathcal{J}[/math], [math]\mathcal{K}[/math] on [ilmath]X[/ilmath] we say:
[math]\mathcal{J}[/math] is finer, larger or stronger than [math]\mathcal{K}[/math] if [math]\mathcal{J}\supset\mathcal{K}[/math]

Larger is a good way to remember this as there are 'more things' in the larger topology.