Pseudometric Topology

A pseudometric topology is the topology on a set whose open sets are the unions of open balls

where is a pseudometric on , is any point of , and (Knapp 2016, p. 84).

A metric topology always makes a T₄-space. A pseudometric topology makes a T₀-space iff implies , in which case is a metric. Indeed, if , every open ball containing contains and conversely, so no open set can separate them. If , the open ball centered at with radius contains but not .

Let send each point to its equivalence class under the equivalence relation iff . The formula

defines a metric on . Its metric topology is the quotient topology on this quotient space. The open sets in are exactly the inverse images of open sets in this metric space (Knapp 2016, p. 95).