Cardinal functions related to quasi-uniformities
by
Attila Losonczi
Eotvos University Budapest

1  

Cardinal functions related to quasi-uniformities

Attila Losonczi   (Budapest)

Let (X, \tau) be a topological space. We are interested in the following type of questions. What is the cardinality of the set composed of all compatible quasi-uniformities? (A quasi-uniformity is said to be compatible with the space if it induces the topology \tau.) What can we say about the cardinality of the set of all compatible transitive or totally bounded or non-transitive quasi-uniformities? How can these cardinal functions be estimated with the help of classical cardinal functions?

We have done the first steps towards the direction of solving these problems and here are some of the answers:

There exist either a unique or at least 2^2\aleph₀ compatible quasi-uniformities. Moreover it can be showed that the same is true for the compatible transitive quasi-uniformities, moreover |\pi(\delta¹)|=1 or >= 2^2\aleph₀, where \delta¹ denotes the finest quasi-proximity compatible with \tau and if \delta is a quasi-proximity then \pi(\delta) is composed of all quasi-uniformities which induce \delta.

A possible generalization of this fact is that if \delta is a compatible quasi-proximity in X  such that V_\delta is transitive then |\pi(\delta)|=1 or >= 2^2\aleph₀. It might be also interesting that how |\pi(\delta⁰)| behaves if \delta⁰ exists (\delta⁰ denotes the coarsest compatible quasi-proximity in X). We managed to prove that in the class of locally compact T₂ spaces |\pi(\delta⁰)|=1 if and only if X is compact or non-Lindelöf. If X is non-compact, Lindelöf then |\pi(\delta⁰)| >= 2^2\aleph₀.

If X is an infinite T₂ space then it admits at least 2^2\aleph₀ non-transitive compatible quasi-uniformities.

The set of real numbers with the standard topology is also examined and it is proved that the cardinality of the set of compatible quasi-uniformities is equal to 2^2\aleph₀ and the cardinality of the transitive totally bounded quasi-uniformities is the same.

Email: losonczi@cs.elte.hu

Date received: June 15, 1998