By I. M. James
This ebook could be considered as a bridge among the research of metric areas and normal topological areas. approximately part the booklet is dedicated to quite little-known effects, a lot of that are released right here for the 1st time. the writer sketches a thought of uniform transformation teams, resulting in the speculation of uniform areas over a base and therefore to the idea of uniform protecting areas.
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Sample text
5) are of th( is Completeness Recall that sequence in a metric is said to be a C a u c h y s e q u e n c e , integer k and completion if space X, satisfy with metric ^ ^ for the condition exists an i n t e g e r m,n > k . 1). t h a t for however, '^m'^n' it Λ filter Thus a s e q u e n c e F in D X F of of the discrete whereas in the c a s e filters satisfy in a s a Cauchy f i l t e r . c o n d i t i o n t o be s a t i s f i e d the satisfies of if the uniform t h e c o n d i t i o n are the t r i v i a l the uniform condition.
F subsets For s u p p o s e , of every finite runs through the f i n i t e F refinement belongs Hence an e n t o u r a g e of Given a each entourage these subsets suppose t h a t exists bounded. Zorn's by D - s m a l l Cauchy. 10). 11). only is it as sequentially that each complete, rather d i f f e r e n t sequentially to the sequence. reasons. elementary For m e t r i c spaces space that whereas complete the property X is canplete the r e a l line IR t h e open i n t e r v a l and s o n o t c c m p l e t e .
Therefore σ*σ G i s convergent, by t h e condition, * and s o h a s an a d h e r e n c e p o i n t . of G G, which t h e r e f o r e But uniform spaces. 17). l6). J Xj of x, where for Xj, also product. each index since π j (x) Xj = χj , j, is structure X Let Y the set ccmplete be a c o m p l e t e u n i f o r m y Y of functions we m e a n , of c o u r s e , of u n i f o r m c o n v e r g e n c e . be a u n i f o r m l y Cauchy i n to Y for φ (x) , Cauchy f i l t e r each p o i n t say. the function and s o subset froTi Φ X IlXj X space.