Topology

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

Comparing topologies

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

Coarser, Smaller, Weaker

Given two topologies $$\mathcal{J}$$ , $$\mathcal{K}$$ on $X$ we say:
$$\mathcal{J}$$ is coarser, 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.

Building new topologies

There are a few common ways to make new topologies from old:

1. Product Given topological spaces $(X,\mathcal{J})$ and $(Y,\mathcal{K})$ there is a topology on $X\times Y$ called "the product topology" (the coarsest topology such that the projections are continuous
2. Quotient Given a topological space $(X,\mathcal{J})$ and an equivalence relation $\sim$ on $X$, we can define the quotient topology on $X$ which we often denote by $\frac{\mathcal{J} }{\sim}$
3. Subspace Given a topological space $(X,\mathcal{J})$ and any $Y\subset X$ then the topology on $X$ can induce the subspace topology on $Y$

Common topologies

Discreet topology

Given a set $X$ the Discreet topology on $X$ is $\mathcal{P}(X)$, that is $(X,\mathcal{P}(X))$ is the discreet topology on $X$ where $\mathcal{P}(X)$ is the power set of $X$.

That is every subset of $X$ is an open set of the topology

Indiscreet Topology

Given a set $X$ the indiscreet topology on $X$ is the topology $(X,\{\emptyset,X\})$