Companions to the Double Sequence Theorem of Nagata
by
H.H. Hung
Concordia University

Given a T₀-space X. A family of subsets on X, indexed by elements of X, say, {V(x): x in X}, is said to be cushioned in another, say, {U(x): x in X}, if, for every \xi in X, there is such a neighbourhood \Omega(\xi) that V(x) \cap \Omega(\xi) = \emptyset whenever \xi not in U(x). We write V(x) << U(x).

We have the following two (necessary and) sufficient conditions for the metrizability of X.

I.

For each n in \omega, there are two such families {U_n(x): x in X}, {V_n(x):x in X} of neighbourhoods that

i)

{x} << V_n(x) << U_n(x), and

ii)

given any neighbourhood \Omega of \xi, there is \nu in \omega such that \xi not in U_\nu(x) whenever x not in \Omega.

II.

For each n in \omega, there are three such families {U_n(x): x in X}, {V_n(x): x in X} and {W_n(x): x in X} of neighbourhoods that

i)

V_n(x) \cap V_n(\xi) = \emptyset if \xi not in U_n(x)

ii)

W_n(x) \cap W_n(\xi) = \emptyset if \xi not in V_n(x)

iii)

{U_n(x): n in \omega} is a neighbourhood base at x.

Date received: April 12, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caae-40.