Dimension Theory of General Spaces by A. R. Pears

By A. R. Pears

An entire and self-contained account of the measurement concept of common topological areas, with specific emphasis at the dimensional homes of non-metrizable areas. It makes the topic available to starting graduate scholars and also will function a reference paintings for common topologists. introductory chapters summarize usual leads to common topology, and canopy fabric on paracompactness and metrization. The crucial definitions of measurement stick to and their basic homes are deduced. Many examples are analysed to teach the various extra striking or pathological features of size thought. anywhere it really is invaluable to take action, proofs are given intimately.

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Since X is a T1-space, F also separates the points of X. 13 that X can be embedded in the Tihonov cube indexed by F. 14 that every subspace of a Tihonov space is a Tihonov space and that the topological product of a family of Tihonov spaces is a Tihonov space. We show finally that each Tihonov space X can be embedded in a Tihonov cube of weight equal to the weight of X. 15 Definition. A universal space for a class g of topological spaces is a space Z which is a member of the class g and has the property that each member of g can be embedded in Z.

It is clear that (d) = (c) and we complete the proof by showing that (c) (a). Let (c) hold and let 0: X --›- Y be a continuous function, where Y is a bicompact space. Let zo be a point of Z and let g be the set of closed sets B in Y such that 0-1-(B) is a zero-set in X such that zo E (0-4(B))-. The set g is non-empty since Y e g. Since (c) holds, Pi has the finite intersection property. Thus r) Bes B is non-empty since Y is compact. Let yo E () Bea /3. If V is a neighbourhood of yo which is a zero-set, then V E R .

The family {A n me,,UA }yer of non-empty open sets of A is locally finite. y A n (-L ey UA. The family {D7},yEr consists of closed sets of X and is locally finite in X. 9 that there exists a locally finite closed covering {EA}AEA of A such that EA c UA n A for each A. For each A in A let FA = Ex U U AE7 D7 . Then for each A, FA is a closed set of X and FA c UA n A. And {FA}AEA is a closed covering of A. Furthermore {FA}AeA is similar to WA n AbeA• For if y e11, then D7 C n Ae7 FA so that me,,FA + 0.

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