We define a real-valued function f on a metric space X to be 1-Lipschitz if, for any points x and y in X, |f(x)-f(y)| <= xy.

We define a real-valued function f on a metric space X to be hyperconical if for any points x and y in X, f(x)+f(y) >= xy.

Define Cl(X), the closure of X, to be the set of all 1-Lipschitz hyperconical real-valued functions on X.

For f,g in Cl(X), define ||f,g|| = sup{|f(x)-g(x)| : x in X}.

Exercise. Prove that (Cl(X),||,||) is a metric space.

By an abuse of notation, X is a subspace of Cl(X). For any x in X, let chi_x(y) = xy for all y in X. Then chi_x is 1-Lipschitz and hyperconical. We embed X in Cl(X) by identifying each x with chi_x, which we call the zero cone at x .

Exercise. Prove that if X has exactly 2 points, which are distance d apart, then Cl(X) is isometric to semi-infinite strip of width d in the Manhattan plane.

Exercise. Let R_n be n-tuples of reals. We give R_n the l_infinity metric, i.e., the distance between x = (x_1, ... x_n) and y = (y_1, ... y_n) is max{|x_i-y_i|}. Prove that if X has cardinality n, Cl(X) is isometric to a subspace of R_n.

Let M be any metric space and X any subspace of M. We define the X-projection to be the function F_X from M to Cl(X) where F_X(y)(x) = xy for each x in X and each y in M.

Closures subject to an inherited property.

Let P be an inherited property of metric spaces, such that a universal P metric space exists. We define the P closure of a P space X as follows.

Define Cl_P(X) to be the subset of Cl(X) which is the image, under the projection F_X, of the a universal P space M which contains X.

Exercise. Prove that the definition of the P closure is independent of the choice of M.