ONT Re: Topology
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Note 29
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| 3. Product and Quotient Spaces
|
| 3.1. Continuous Functions (cont.)
|
| 1. Theorem. If X and Y are topological spaces
| and f is a function on X to Y, then the
| following statements are equivalent.
|
| a. The function f is continuous.
|
| b. The inverse of each closed set is closed.
|
| c. The inverse of each member of a subbase for
| the topology for Y is open.
|
| d. For each x in X the inverse of every neighborhood
| of f(x) is a neighborhood of x.
|
| e. For each x in X and each neighborhood U of f(x)
| there is a neighborhood V of x such that f[V] c U.
|
| f. For each net S (or {S_n : n in D}) in X which converges
| to a point s, the composition f o S ({f(S_n) : n in D})
| converges to f(s).
|
| g. For each subset A of X the image of the closure is a subset
| of the closure of the image; that is, f[A˜] c f[A]˜.
|
| h. For each subset B of Y, f^(-1)[B]˜ c f^(-1)[B˜].
|
| Proof.
|
| (a <=> b). This is a simple consequence of the fact that the
| inverse of a function preserves relative complements; that is,
| f^(-1)[Y ~ B] = X ~ f^(-1)[B] for every subset B of Y.
|
| (a <=> c). If f is continuous then the inverse of a member of a subbase is
| open because each subbase member is open. Conversely, since each open set
| V in Y is the union of finite intersections of subbase members, f^(-1)[V]
| is the union of finite intersections of the inverses of subbase members;
| if these are open, then the inverse of each open set is open.
|
| (a => d). If f is continuous, x in X, and V is a neighborhood of f(x),
| then V contains an open neighborhood W of f(x) and f^(-1)[W] is an open
| neighborhood of x which is a subset of f^(-1)[V]; consequently f^(-1)[V]
| is a neighborhood of x.
|
| (d => e). Assuming (d), if U is a neighborhood of f(x), then
| f^(-1)[U] is a neighborhood of x such that f[f^(-1)[U]] c U.
|
| (e => f). Assuming (e), let S be a net in X which converges to a point s.
| Then if U is a neighborhood of f(s) there is a neighborhood V of s such that
| f[V] c U, and since S is eventually in V, f o S is eventually in U.
|
| (f => g). Assuming (f), let A be a subset of X and s a point of the closure A˜.
| Then there is a net S in A which converges to s, and f o S converges to f(s),
| which is therefore a member of f[A]˜. Hence f[A˜] c f[A]˜.
|
| (g => h). Assuming (g), if A = f^(-1)[B], then f[A˜] c f[A]˜ c B˜
| and hence A˜ c f^(-1)[B˜]. That is, f^(-1)[B]˜ c f^(-1)[B˜].
|
| (h => b). Assuming (h), if B is a closed subset of Y,
| then f^(-1)[B]˜ c f^(-1)[B˜] = f^(-1)[B] and f^(-1)[B]
| is therefore closed.
|
| þ
|
| There is also a localized form of continuity which is useful. *
| A function f on a topological space X to a topological space Y
| is 'continuous at a point' x iff the inverse under f of each
| neighborhood of f(x) is a neighborhood of x. It is easy to
| give characterizations of the form of 3.1.e and 3.1.f for
| continuity at a point. Evidently f is continuous iff
| it is continuous at each point of its domain.
|
|
| * If f is defined on a subset A of a topological space,
| then continuity at points of the closure A˜ may also be
| defined (see 3.D); several useful propositions result.
|
| JLK, Gen Top, pages 86-87.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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