SOME MAPPING THEOREMS
SOME MAPPING THEOREMS
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DOI：
10.1111/j.2164-0947.1971.tb02598.x
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发表时间：
1971-03
影响因子：
5.2
通讯作者：
Vincent J. Mancuso
中科院分区：
文献类型：
--
作者：
Vincent J. Mancuso
In this paper, a space will mean a regular T, space and a mapping will mean a continuous surjection. Some preliminary definitions are in order. A mapping f: X+ Y is open (closed) if f maps open (closed) subsets of X onto open (closed) subsets of Y. A mapping f is finite-to-one if f-*(y) is a finite subset of X for each y in Y. A mapping f is perfect if f is a closed mapping and fl (y) is a compact subset of X for each y in Y. Let ip be a topological property and f: X+ Y a mapping. We will be concerned with the following question: If Y has F will X have F? Note that for F=“compactness” our basic question will have a negative answer even if Y is a compact metrizable space and f is an open? nd closed mapping. This can be seen by simply mapping a noncompact space onto a point. This trivial example shows that for a given property? we should not expect an affirmative answer to our question unless some rather severe restrictions are placed either on the mapping or on the domain. It is very surprising that this fundamental question was not extensively studied until as recently as 1966, when Proizvolovl proved that if X is a locally compact space and f: X+ Y is an open, finite-toone mapping, then the weight of X (ie, the least cardinal of an open base of X) does not exceed the weight of Y. Also in 1966, Arhangel’skii2 showed that if f: X+ Y is an open, closed, and finite-to-one mapping, then the weight of X does not exceed the weight of Y. In particular, under the hypotheses of each of these theorems, X is second-countable whenever Y is second-countable.
