ONT Re: Topology
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| 1. Topological Spaces
|
| 1.1. Topologies and Neighborhoods (cont.)
|
| The discrete and the indiscrete topology for a set X are
| respectively the largest and the smallest topology for X.
| That is, every topology for X is contained in the discrete
| topology and contains the indiscrete topology. If !T! and
| !U! are topologies for X, then, following the convention
| for arbitrary families of sets, !T! is smaller than !U!
| and !U! is larger than !T! iff !T! c !U!. In other words,
| !T! is smaller than !U! iff each !T!-open set is !U!-open.
| In this case it is also said that !T! is 'coarser' than !U!
| and !U! is 'finer' than !T!. (Unfortunately, this situation
| is described in the literature by both of the statements:
| !T! is 'stronger' than !U! and !T! is 'weaker' than !U!.)
| If !T! and !U! are arbitrary topologies for X it may happen
| that !T! is neither larger nor smaller than !U!; in this
| case, following the usage for partial orderings, it is
| said that !T! and !U! are not 'comparable'.
|
| JLK, Gen Top, pages 37-38.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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