ONT Re: Topology
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Note 28
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| 3. Product and Quotient Spaces
|
| 3.1. Continuous Functions (cont.)
|
| A map f of a topological space (X, !T!) into a topological space (Y, !U!)
| is 'continuous' iff the inverse of each open set is open. More precisely,
| f is continuous with respect to !T! and !U!, or (!T!, !U!)-continuous, iff
| f^(-1)[U] is in !T! for each U in !U!. The concept depends on the topology
| of both the range and the domain space, but we follow the usual practice
| of suppressing all mention of the topologies when confusion is unlikely.
|
| There are one or two propositions about continuity which are
| quite important, although almost self-evident. First, if f is
| a continuous function on X to Y and g is a continuous function
| on Y to Z, then the composition g o f is a continuous function
| on X to Z, for (g o f)^(-1)[V] = f^(-1)[g^(-1)[V]] for each
| subset V of Z, and using first the continuity of g, then that
| of f, it follows that if V is open so is (g o f)^(-1)[V].
|
| If f is a continuous function on X to Y, and A is a subset
| of X, then the restriction of f to A, f|A, is also continuous
| with respect to the relative topology for A, for if U is open
| in Y, then (f|A)^(-1)[U] = A |^| f^(-1)[U], which is open in A.
| A function f such that f|A is continuous is 'continuous on' A.
| It may also happen that f is continuous on A but fails to be
| continuous on X.
|
| JLK, Gen Top, pages 85-86.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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