Yes, topologies is a Scrabble word!

Noun

1. Topographic study of a given place (especially the history of the place as indicated by its topography)

2. The study of anatomy based on regions or divisions of the body and emphasizing the relations between various structures (muscles and nerves and arteries etc.) in that region

3. The branch of pure mathematics that deals only with the properties of a figure X that hold for every figure into which X can be transformed with a one-to-one correspondence that is continuous in both directions

4. The configuration of a communication network