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By Iain T. Adamson

This paintings goals to provide common topology in an unconventional means. It provides a evaluate of the elemental definitions including routines with out recommendations or proofs of the theorems partially 1, after which offers the suggestions partly 2, permitting the coed to match solutions with their very own.

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Example text

Suppose C is closed in the union UD n , where each D.. is perfectly normal and closed in X. First let f1 be any nonnegative continuous function on D1 zero exactly on C nD1. I i

Then ((,(B l ) CXn-l(l(X -A), and fJ(B l ) C p(Xn-l(l(X -A)). Also fJ(B 2 ) =p«({'(En)(lA) =f«({,(En)(lA) C Yn-l becausef is cellular. Thus fJ(En) is contained in a union of cells of dimension less than n. If fJ = pif;, then if; (En) C Y n-l because (Y, 'lr) is a cell structure. This completes the first part of the theorem. The statement about strict equivalence is very easy and will be omitted. 2. We denote the cell complex obtained from cell complexes (X,s) and (Y,::I) by adjunction of X to Y as above by (YV, X, SV,::I) and call it the adjunction complex.

2 (1) each T i is closed in X, and the finite union Ui T i is closed in X. Thus crnA = crn (U i Ti) is closed in cr. Since X has the weak topology with respect to s, A is closed in X. Since the subspace A CX is Hausdorff and properties (i) and (iii) are obviously inherited by the sub complex (A,::I), to prove (2) we need only check property (ii) for (A,::I). If A is given the relative topology and F is closed in A, then F is closed in X because, by (1), A is closed. But then F ncr is closed in cr for each cr E S, and, in particular, F ncr is closed in cr for each cr E::I.

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