By I.M. James

Scholars of topology rightly whinge that a lot of the fundamental fabric within the topic can't simply be present in the literature, a minimum of no longer in a handy shape. during this publication i've got attempted to take a clean examine a few of this easy fabric and to arrange it in a coherent style. The textual content is as self-contained as i'll kind of make it and will be really available to somebody who has an straight forward wisdom of point-set topology and workforce concept. This publication is predicated on a process sixteen graduate lectures given at Oxford and in different places at times. In a process that size one can't speak about too many subject matters with out being unduly superficial. notwithstanding, this used to be by no means meant as a treatise at the topic yet relatively as a brief introductory direction for you to, i'm hoping, end up helpful to experts and non-specialists alike. The advent incorporates a description of the contents. No algebraic or differen tial topology is concerned, even supposing i've got borne in brain the wishes of scholars of these branches of the topic. workouts for the reader are scattered in the course of the textual content, whereas feedback for extra studying are inside the lists of references on the finish of every bankruptcy. often those lists comprise the most assets i've got drawn on, yet this isn't the kind of publication the place it really is doable to provide a reference for every little thing.

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1. For each X and each a, n w ( C a ( X ) ) _< w(X). Proof. Let base for C(X). ~. 8 be a base for X such t h a t T h e family {[B,V]: B • To see t h a t that a • Ba c Bal O ... O contains f w(%(x))~, is ~, • ,v, = w(X). For in [A,V]. that I /3 I = nw(Ca(X)). 3r = a-network anT(X), reverse {F: and for 3r}. 3r such t h a t show t h a t F* c U. ,a n • open = t3 such A such t h a t subset of nw(C (X)) _< A c C a ( X ) which nw(Cr(X)) < let V be • a countable base finite i n t e r s e c t i o n s for 7, let jr define To show 0.

Weight. for X provided Then A collection i~ of n o n e m p t y open s u b s e t s of a space X is called a z r - b a s e the ~r-base that rr-weight for X}. every n o n e m p t y open of is X Also the defined Proof. It a-network of X contains be lrw(X) of X = is w some member of + given by min(IBl: B aanw(X) = B. is a w + on X}. For each X and each a , a a n w ( X ) = a a ( X ) - a n w ( X ) . 1. subset an from a-cover, definitions then that aa(X) anw(X) _< _< a a n w ( X ) . aanw(X). anw(X) _< aanw(X).

Then Direct the set (fi(xj))(i,j)EixJ is a The net (fi) converges continuously to f for e v e r y x E X and e v e r y net (x j) in X converging to x in X, the net (fi(xj)) converges to f(x) in R. 5. 2. A topology r on C(X) is a splitting topology if and only if w h e n e v e r (fi) is a net in C(X) converging continuously to f E C(X) then (fi) converges to f in C r ( X ). Proof. First suppose that C(X) converging continuously where r is a splitting to f E C(X). Define the space that ( i } for some i0 E I.