# space of metric spaces of a topological space

Given a topological space that is T3 (i.e. regular) and has a sigma-discrete base (where these conditions allow the topological space to be biconditionally metrizable), is there a way to understand what all the possible admittable metrics are? And do all these possibly admittable metrics form a space?

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The space of metrics is just the space of continuous functions $X \times X \to \mathbb{R}$ satisfying certain properties and inducing the correct topology on $X$, so it can be given the subspace topology inherited from the compact-open topology. Do you have an intended application? – Qiaochu Yuan Feb 17 '11 at 14:45
I just wanted to know whether or not such spaces had been studied, and if there is a sophisticated way of analysing them. – user7485 Feb 17 '11 at 14:52