ONT Topology
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| 1. Topological Spaces
|
| 1.1. Topologies and Neighborhoods
|
| A 'topology' is a family !T! of sets which satisfies the two conditions:
| the intersection of any two members of !T! is a member of !T!, and the
| union of the members of each subfamily of !T! is a member of !T!. The
| set X = |_|{U : U in !T!} is necessarily a member of !T! because !T!
| is a subfamily of itself, and every member of !T! is a subset of X.
| The set X is called the 'space' of the topology !T! and !T! is a
| 'topology for X'. The pair (X, !T!) is a 'topological space'.
| When no confusion seems possible we may forget to mention the
| topology and write "X is a topological space". We shall be
| explicit in cases where precision is necessary (for example
| if we are considering two different topologies for the same
| set X).
|
| The members of the topology !T! are called 'open' relative to !T!, or
| !T!-open, or if only one topology is under consideration, simply open
| sets. The space X of the topology is always open, and the void set is
| always open because it is the union of the members of the void family.
| These may be the only open sets, for the family whose only members are
| X and the void set is a topology for X. This is not a very interesting
| topology, but it occurs frequently enough to deserve a name; it is
| called the 'indiscrete' (or 'trivial') topology for X, and (X, !T!)
| is then an 'indiscrete topological space. At the other extreme is
| the family of all subsets of X, which is the 'discrete' topology
| for X (then (X, !T!) is a 'discrete topological space'). If !T!
| is the discrete topology, then every subset of the space is open.
|
| JLK, Gen Top, page 37.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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