Separation Pseudocharacter and Cardinality of Topological Spaces
by
Dimitrina N. Stavrova
Miami University

It is well known that in T₁-topological space X a cardinal invariant called pseudocharacter can be introduced in a natural way - we say that \Psi(X) <= \tau iff for every x in X there is a neighborhood system H(x)={ U_\alpha(x) :\alpha in \tau} such that {x} = \cap H(x) for every x in X.

In [4] R. Hodel introduced a similar invariant for Hausdorff space X - we say that the Hausdorff pseudocharacter H\Psi(X) <= \tau iff for every x in X there is a neighborhood system H(x) = { U_\alpha(x) : \alpha in \tau} such that for every two points x =/= y there are U_\alpha(x) in H(x) and U_\alpha(y) in H(y) such that U_\alpha(x) \cap U_\alpha(y) = \emptyset. He also proved that \Psi(x) <= H\Psi(X) <= \chi(X) and gave examples that these inequalities can be strict. He strengthened the main theorems in the theory of cardinal invariants by replacing \chi(X) with H\Psi(X) in the Hausdorff case.

Here in an Urysohn space X we introduce similar invariant depending on separation axiom - we say that the Urysohn pseudocharacter of X - U\Psi(X) <= \tau iff for every x in X there is a neighborhood system H(x)={ U_\alpha(x): \alpha in \tau} such that for every two points x =/= y there are U_\alpha(x) in H(x) and U_\alpha(y) in H(y) such that [`(U<sub>\alpha</sub>(x))] \cap [`(U_\alpha(x))]=\emptyset. It is clear that \Psi(x) <= H\Psi(X) <= U\Psi(X) <= \chi(X).

In [2] U.N.B. Dissanayeke and S. Willard introduced the almost Lindelöf number for X as aL(X) = \omega·min{\tau: for every open cover \gamma of X there is a \gamma' in [\gamma] ^{<= \tau} such that  \cup {[`U] :U in \gamma'} = X}. In [1] A. Bella and F. Cammaroto proved that for Urysohn spaces we have that |X| <= 2^{aL(X, X)·\chi(X)}. We strengthen this result to :

Theorem 1. If X is Urysohn space then |X| <= 2^{aL(X, X)·U\Psi(X)}.

In [3] A.Gryzlov and D. Stavrova proved that for a given subset X₀ of a Hausdorff space X we have that |X\X₀| <= 2^{L(X, X₀)·\Psi_C(X)·t(X)} where L(X, X₀) = \omega·min{\tau: for every open cover \gamma of X there is a \gamma' in [\gamma] ^{<= \tau} such that  X\X₀ subset or equal \cup \gamma'}. Here a similar result using H\Psi(X) is given:

Theorem 2. If X is Hausdorff and X₀ subset or equal X then |X\X₀| <= 2^{L(X, X₀)·H\Psi(X)}.

REFERENCES

1. Bella A., Cammaroto F., On the cardinality of Urysohn spaces, Canad. Math. Bull., 31 (1988), 153-158

2. Dissanayeke U.N.B., Willard S., The almost Lindelöf degree, Canad. Math. Bull., 27 (1984), 452-455.

3. Gryzlov A.A., Stavrova D.N., Topological spaces with a selected subset - cardinal invariants and inequalities, Comptes rendue de l'Academie Bulgare de Sciences, vol. 46, no. 7, 1993, 17-19.

4. Hodel R.E., Combinatorial set theory and cardinal function inequalities, Proceedings of the AMS, vol. 111, no. 2, 1991, 567-573.

Date received: June 22, 2000

Copyright © 2000 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 # caeu-41.