By I. M. James

This booklet should be considered as a bridge among the examine of metric areas and common topological areas. approximately part the ebook is dedicated to fairly little-known effects, a lot of that are released the following for the 1st time. the writer sketches a thought of uniform transformation teams, resulting in the speculation of uniform areas over a base and as a result to the idea of uniform masking areas.

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The s u b s e t of such t h a t where so non-empty uniforml; of points open s e t s i s dense in Η Proposition a χ e D[H], number of v a l u e s o f Therefore this of ζ = (ξ^) number o f v a l u e s of is is . be a f a m i l y most a f i n i t e subsets, Δ Then t h e u n i f o r m p r o d u c t a t most a f i n i t e is uniformly is Now since χ « Η Let of but a f i n i t e indexing H, λ connected. Η nUj c nXj So i f spaces. this, subset D. 9). is uniformly connected. a s y m m e t r i c e n t o u r a g e of s y m m e t r i c e n t o u r a g e of for i s uniformly and s o Let φ : X -·• Y are uniform uniform spaces.

S on is X, F' such t h a t F We d e d u c e e a c h Cauchy s e q u e n c e S X each f i n i t e F finite each u l t r a f i l t e r on t h e m e t h o d u s e d of the corresponding X - DLT] The u n i f o r m line subspace is is C h o o s e a D - s m a l l member for i s complete. that F', X Thus we o b t a i n o u r c o n t r a d i c t i o n example the r e a l the F. (the refinement. be a p o i n t and G. 9) . 8) Recall cOTplete is and l e t Τ = S υ {χ}. Since to obtain a c o n t r a d i c t i o n , G D not t o t a l l y a filter a Cauchy r e f i n e m e n t F Since to filter Cauchy.

And β - Xj^^j^r and β ζ c a n be j o i n e d t o is necessary - 1. ^nd α For can be can be j o i n e d t o η by by a D - c h a i n , to take c a r e , and c o i n d u c e d u n i f o r m as in general, structures, this i s not we h a v e i m p o s e d . Proposition be a c o m p a t i b l e e q u i v a l e n c e Let R t i o n on t h e u n i f o r m space X. 19). A sense that S = R η (AxA) on the j. Let {Xj} be a f a m i l y Then t h e r e l a t i o n uniform product s t r u c t u r e , SjRjHj for each index The p r o o f s the reader .