By Symposium in Pure Mathematics Stanford University 1976, Visit Amazon's R. James Milgram Page, search results, Learn about Author Central, R. James Milgram, , American Mathematical Society
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Extra resources for Algebraic and geometric topology
If Ε i s any e n t o u r a g e The uniform topology of Y then D = (φ^φ) ^E i s an e n t o u r a g e o f X such that false, as φΟ[Η] c Ε[φΗ] Therefore φΗ c φΗ, However, as t h e c o n v e r s e of every undergraduate X = Y =JR, required. student the r e a l f u n c t i o n g i v e n by line, φ(t) uniformly continuous, Similarly for therefore, functions. Under t h e topological equivalences. 11). Let D[xJ φ '•R i s uniformly that φ(x) Let R . i s defined from continuous and continuous equivalences become uniform t o t h e c a t e g o r y of topolog- a s shown by φ : X Y be u n i f o n n l y Then where for Κ c γ φ and each entourage open t h e r e e x i s t s and s o I χ Μ frcra t h e c a t e g o r y of spaces.
E. terminates. of X are the indices of t with each such , with i^ i for < i2 those We t as which . i^^ , then E. = D. t i η l «D. »D. 1, 1 E^ = ΔΧ , if s,t ε I' the diagonal. with s < t then E^ c e ^ . The uniform topology This is o b v i o u s when (k+l)/2n , a r e of case now f o l l o w s t h e form for We now d e f i n e = infit X , ς i EgCxJ remains In f a c t , ζ < s α is α(ξ) if CK/2" , a few s l i p s . in s t The g e n e r a l . index i numerator in a^ . Κ by If ζ ε Η α(ς) = l then .
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 . of if (and ccmplete. exeimple, in the Euclidean metric, fore, for are always s p a c e s are a l s o The m e t r i c for if in sequentially T h i s shows, not i s complete s i n c e we c a n a p p l y t h e c o n d i t i o n associated if) X spaces spaces, Obviously c o n p l e t e uniform canplete, space convergent.