Thinking out loud here, but metric spaces are examples of topological spaces... so is there an equivalent "generalised topological space" for Lawvere metric spaces?

I can see how dropping axiom (d) just means the space has indistinguishable points (ie it no longer has to be \\(T_0\\)), and dropping (f) just means it no longer has to be connected... but dropping axiom (e) seems to have stranger effects, ones that can't obviously be captured by our usual concept of topology.

I can see how dropping axiom (d) just means the space has indistinguishable points (ie it no longer has to be \\(T_0\\)), and dropping (f) just means it no longer has to be connected... but dropping axiom (e) seems to have stranger effects, ones that can't obviously be captured by our usual concept of topology.