By Marvin J. Greenberg

ISBN-10: 0805335579

ISBN-13: 9780805335576

ISBN-10: 0805335587

ISBN-13: 9780805335583

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1 M Definition c PX. (Choquet [ 3 ] ) : Let Then M is called a grill X be a set and l e t on X i f f the following three axioms are s a t i s f i e d : G: A e M and A c B C X implies G2: A u B e M implies A e M or Bell. Be l l , G 3 : <> f 4 M. 2 Definition: be a g r i l l on Let X. (X,^) be a nearness space and l e t M Then M is called a z-bunch i f f the following axioms are s a t i s f i e d : c B^: M X and Cl M e M implies B2: M e £ and M t 4. 51 M e M. H. L. BENTLEY The concept of a £-bunch is a natural analogue of what has been called a bunch in a Lodato proximity space.

1 M Definition c PX. (Choquet [ 3 ] ) : Let Then M is called a grill X be a set and l e t on X i f f the following three axioms are s a t i s f i e d : G: A e M and A c B C X implies G2: A u B e M implies A e M or Bell. Be l l , G 3 : <> f 4 M. 2 Definition: be a g r i l l on Let X. (X,^) be a nearness space and l e t M Then M is called a z-bunch i f f the following axioms are s a t i s f i e d : c B^: M X and Cl M e M implies B2: M e £ and M t 4. 51 M e M. H. L. BENTLEY The concept of a £-bunch is a natural analogue of what has been called a bunch in a Lodato proximity space.

Similarly X- Y . THEOREM. 3 iM>= lw l X and Y are locally compact metric spaces of which one is compact and the other is not (or one is separable are and the other is not)> then Sh (X) and Sh (Y) incomparable. Proof. 2, X and Y have proper neighborhood systems metric spaces If U(Y,Q) Y l/(X,Q), ti(Y,Q) in some locally compact P, Q containing X and Y, respectively. is compact and X is not, then some member of is compact but no member of U(X,P) is compact. Since a proper map with noncompact domain cannot have a compact range, there is no proper mutation from 22 K(X,P) to STUDIES IN TOPOLOGY (Y,Q), and hence Sh^(X) and Sh^(Y) are incomparable.

### Algebraic topology: a first course by Marvin J. Greenberg

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