Conventions

Separation axioms

For the separation axioms, T_n ⇒ T_m whenever n ≥ m. For example regular is defined to assert that closed points and sets can be separated; T₃ is defined to be both regular and T₀. See e.g. wikipedia for more information.

Local Properties

If a property is named "locally P", then that means that every point in the space has a neighborhood base satisfying P for every member of the base. On the other hand, some authors define "locally P" to mean there is a single neighborhood satisfying P for each point. These definitions are occasionally equivalent (e.g. locally metrizable), but are not equivalent in general (e.g. locally compact). See this issue for discussion.

Use "locally P" when there's a basis of P neighborhoods, and (when not equivalent) use "weakly locally P" when there's a single P neighborhood.