On topological spaces with a unique compatible quasi-uniformity

Abstract

It is shown in [2] that a uqu space satisfies the following conditions. {DC) There is no infinite, strictly decreasing sequence of open sets with open intersection. (IC) There is no infinite, strictly increasing sequence of open sets.In this note we show that for a transitive space these conditions are sufficient for the space to be uqu. This will follow as a consequence of the following result. THEOREM 1. Let y be a complete lattice of sets under the operations of intersection and union, in which all chains are finite. Then Sf is finite.Proof. This result is a consequence of standard theorems on lattice theory. See for example Corollary 2 of [1, p. 59] or Theorem 31 of [3, p. 116]. However, for the sake of completeness we give a direct proof. Since SP is a complete lattice we have for all ^e A^ u { S | S e y } . Also, for SeS?, 5 = v{A x \xeS} and so it will suffice that the number of distinct sets A x (xeX) is finite. We will call A a minimal\f xeA a implies A x = A a . Clearly distinct minimal sets are disjoint. Since all chains in £P are finite we see there are just finitely many distinct minimal sets, say A ai , ..., A Bn , that there exist a finite number of sets A bl , ..., A bm , which cover X, and that for any xeX there exist i,j with A ai £ A x £ A bj . Hence we may complete the proof by showing that if A a c A b then there are just finitely many distinct A x with A a £ A x £ A b . We will write A u < A v if A v is an immediate successor of A u for the ordering £ . Given A x c A y there exists zeX with A x < A Z Q A y , and hence there is a finite sequence A a < A., < ... < A Zt < A b , since £f contains no infinite chains. Moreover, for the same reason, the number of terms in such a sequence is bounded and each A x can only have a finite number of distinct immediate successors A z , so there are only finitely many distinct sequences A a < A 2l < ... < A, t < A b . Finally since each A x with A a c A X Q A b belongs to one of these sequences the proof is complete.An examination of the above proof shows that we have actually established the slightly more general result that if Sf is a set of sets, and if 9", the set of finite unions of the sets A x (xeX) is a subset of £f and contains no infinite chains, then y is finite.We return now to the question of uqu spaces. We recall that an open cover ( H> of the topological space (X, ST) is called a Q-cover if for all xeX. The quasi-uniformity <^Q with subbase is a Q-cover for (X,3T)} Glasgow Math. J. 18 (1977) 11-12.

Topological Spaces with a Unique Compatible Quasi-Uniformity

“…Let U E Fine($). Then for each nELN there is a U n E Fine($) such that U 2 n + X C U n and U\ C U. We show that U belongs to the fine-transitive quasi-uniformity for X.…”

“…[6], Lemma 3). In [2] it was essentially proved that the following two conditions are equivalent for a topological space X.…”

“…The first part of Problem B is answered negatively in [7]; in this note we answer the second part positively. Topological spaces that admit a unique quasi-uniformity are considered in [1][2][3][4][5][8][9][10]. Related questions are dealt with in [6,7].…”