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 $(X,\mathcal{J})$ and $(X,\mathcal{K})$ be two topologies on $X$

Coaser, Smaller, Weaker

Given two topologies

$$\mathcal{J}$$ , $$\mathcal{K}$$ on $X$ we say:
$$\mathcal{J}$$ is coaser, smaller or weaker than $$\mathcal{K}$$ if $$\mathcal{J}\subset\mathcal{K}$$

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

Finer, Larger, Stronger

Given two topologies

$$\mathcal{J}$$ , $$\mathcal{K}$$ on $X$ we say:
$$\mathcal{J}$$ is finer, larger or stronger than $$\mathcal{K}$$ if $$\mathcal{J}\supset\mathcal{K}$$

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