New PDF release: Applicable Geometry (1977)(en)(207s)

By Heinrich W Guggenheimer

ISBN-10: 0882753681

ISBN-13: 9780882753683

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Additional info for Applicable Geometry (1977)(en)(207s)

Example text

We show the equivalence, in T1 spaces, of the two forms of its definition used in analysis. 2 Given a T1 space (X, T ), p ∈ X and A ⊆ X, the following are equivalent: (i) Every neighbourhood of p contains infinitely many points of A (ii) Every neighbourhood of p contains at least one point of A different from p. Proof Obviously, (i) ⇒ (ii); conversely, suppose (i) fails; so there exists a neighbourhood N of p such that N ∩ A is finite. Consider H = [X \ (N ∩ A)]∪{p}; it is cofinite and is thus an (open) neighbourhood of p.

E. C is the weakest of all the T1 topologies that can be defined on X. Proof is left to the reader. ). We show the equivalence, in T1 spaces, of the two forms of its definition used in analysis. 2 Given a T1 space (X, T ), p ∈ X and A ⊆ X, the following are equivalent: (i) Every neighbourhood of p contains infinitely many points of A (ii) Every neighbourhood of p contains at least one point of A different from p. Proof Obviously, (i) ⇒ (ii); conversely, suppose (i) fails; so there exists a neighbourhood N of p such that N ∩ A is finite.

We expected a definition like ‘subsequence’ to turn up here and we are disappointed that it has to be so complicated. Net theory ceases to be a straightforward generalisation of sequence theory precisely when we have to take a subnet . . so we’ll try to avoid this whenever possible! 7 (X, T ) is compact iff in X, every net has (at least one) convergent subnet. ) Proof Not required. 1 Compactness is closed-hereditary 31 Proof (for if (xα ) is a net in a closed set F ⊆ X, then it has a convergent subnet (yβ ) in X.

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Applicable Geometry (1977)(en)(207s) by Heinrich W Guggenheimer


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