Topology

Read e-book online Algebraic Topology: A Primer (Texts and Readings in PDF

By Satya Deo

ISBN-10: 8185931682

ISBN-13: 9788185931685

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Extra info for Algebraic Topology: A Primer (Texts and Readings in Mathematics)

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Therefore lub{Fi} is the least fine topology on X which is at least as fine as the topologies Fi ( i E I ) . This is the least upper bound of the topologies Ti ( i E I). Fi}, This is the finest topology on X which is at least as coarse as any of the topologies Ti . It is called the greatest lower bound of the topologies Ti ( i E I ) . < mi EX E RClSES 1. Let Tx be the topology on X whose open sets are X and all sets not containing the point x E X. Determine lub(9, : x E X} and glb{Fz : x E X}. 5 .

It is clear that the family O(x) of open neighborhoods 0, of a point x is a filter base and the filter generated by O(x) is J l r ( x ) . T h e closed neighborhoods of x form a family V(x) which is a filter base but the filter generated by V ( x ) is not necessarily N ( x ) . This is the case for instance if X is infinite and its topology is the topology of finite complements. Then the only closed neighborhood of any point x is the entire space X . T h e problem which we discuss here is twofold: First suppose that the neighborhood filter =,V(x) of each point x of a topological space X is known.

Definition 2. A point x E X is a boundary point of the set A if every open set 0, containing x intersects both A and c A . The set of boundary points is called the boundary of A and is denoted by Ah. It is obvious that Ab = ( c A ) ~Moreover, . x is a boundary point of a set A if and only if x 4 Ai u Ae. Therefore the boundary A b is the complement of A' u Ae. Since A' and Ae are disjoint we obtain: For every set A of a topological space X its interior A', exterior Ae, and boundary Ah are disjoint sets whose union is X .

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Algebraic Topology: A Primer (Texts and Readings in Mathematics) by Satya Deo


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