Foundations of topology: an approach to convenient topology by Gerhard Preuß

By Gerhard Preuß

A brand new beginning of Topology, summarized less than the identify handy Topology, is taken into account such that numerous deficiencies of topological and uniform areas are remedied. this doesn't suggest that those areas are superfluous. It ability precisely higher framework for dealing with difficulties of a topological nature is used. during this environment semiuniform convergence areas play a vital function. They contain not just convergence constructions comparable to topological constructions and restrict area constructions, but in addition uniform convergence buildings resembling uniform buildings and uniform restrict house constructions, and they're compatible for learning continuity, Cauchy continuity and uniform continuity in addition to convergence buildings in functionality areas, e.g. basic convergence, non-stop convergence and uniform convergence. a number of attention-grabbing effects are offered which can't be bought through the use of topological or uniform areas within the ordinary context. The textual content is self-contained except for the final bankruptcy, the place the intuitive idea of nearness is included in handy Topology (there already exist first-class expositions on nearness spaces).

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Reg] (resp. ) @ Th e category L Con (resp . LPCon) of locally connected (resp. locally pathconnected) topological spaces [and continuous maps ]. (Hint. 1 Special categorical properties of topological constructs Completeness and cocompleteness 1. 1. 1 T h eor em. Let C be a construct. 2. 2. 35 (b) For any set X , any family ((Xi , ~i) )iEl of C- objecis indexed by some class I and any family (Ii : Xi ----+ X)iEl of maps indexed by I there exists a unique C- siruciure C. i) , I i, X , 1) , i. e.

A m onom orphism iff f : X --* Y 'is injective. I b) an epimo rphism iff f : X ---t Y is surjectiv e. P ro of a) ex) Let x , y E X such that f (x) = f(y) · x : (X , ~rI) --* (X, O defined by x (z) = x for each z E X and y : (X, ~rI) ---t ( X ,~) defined y(z) = y for each z E X are C-morphisms (cf. 2 2)) such th at f o x = I 0 y. e. x = y. Sinc e I is injective 1(X' ) = 8(x') for each x' E X' . Thus 1 = 8. (X,~) be Cmorphisms such th a t f 0 1 = f 0 8. 2. 5 Theorem. b) a) (indirect) . Suppose t hat f is not surjective.

JX;))iEI a family of semiuniform convergence spac es, (j; : X ---+ X i)iEI a family of maps, then JX = {F E F(X x X) : (j; x j;)(F) E J X, for each i E I} is th e initial SUConv-structure on X with resp ect to th e given data. e. such t hat A sati sfies the following condit ions: 1) X E A, 2) A E A implies X\A E A , 3) UnEIN An E A whenever (An)nElN is a sequence in At» A map f : (X , A) ---+ (X' , A') between measur abl e spaces is called measurable provided th at f - I[A' ] E A for each A' E A' .

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