The Behavior of Orthocompactness in Products, with Emphasis on Certain Analogies with that of Normality Under Similar Circumstances, Or Orthocompact: Metacompact - Normal - Paracompact (a.E.) |
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Page 12
... sets is a rank 1 collection iff , whenever A , B & F , then either A B , BC A , or An B = Ø . ɛ Definition 1.0.1 . A space is non - Archimedean iff it has a rank 1 base of open sets . ( Thus , for example , the Cantor Space is non ...
... sets is a rank 1 collection iff , whenever A , B & F , then either A B , BC A , or An B = Ø . ɛ Definition 1.0.1 . A space is non - Archimedean iff it has a rank 1 base of open sets . ( Thus , for example , the Cantor Space is non ...
Page 31
... open sets at y . α Let U be any open cover of X x Y. For each < x , y > ε X x Y there are an open nbhd V , ( x ) of x , an a ( x ) ε к , and a U ɛ U such y that < x , y > ɛ V ̧ , ( x ) x B Ba ( x ) ( y ) ≤ y U. Fix yε Y , and consider ...
... open sets at y . α Let U be any open cover of X x Y. For each < x , y > ε X x Y there are an open nbhd V , ( x ) of x , an a ( x ) ε к , and a U ɛ U such y that < x , y > ɛ V ̧ , ( x ) x B Ba ( x ) ( y ) ≤ y U. Fix yε Y , and consider ...
Page 69
... open subset of X is orthocompact . But an open subset of a LOTS can of course be written as a disjoint union of convex open sets , each of which is a LOTS in its subspace topology and therefore ortho- compact , and the result follows ...
... open subset of X is orthocompact . But an open subset of a LOTS can of course be written as a disjoint union of convex open sets , each of which is a LOTS in its subspace topology and therefore ortho- compact , and the result follows ...
Common terms and phrases
a x ß A-System analogy assume base box products cf(a cf(B cf(k Chapter clearly clopen closed copy closed subset cofinal collectionwise normal compact space construct contains a closed Corollary countably compact countably metacompact CSEP D₂ define Definition discrete union Dowker spaces Example F is s.d. finite guarantees Hausdorff Hausdorff space hereditarily orthocompact homeomorphic infinite products Kunen Lemma Lemma 7.0.6 holds let H Morita's non-Archimedean numbers obviously open cover open nbhd open refinement open sets open subset ord(x ortho orthocompact iff orthocompact space orthocompactness and normality P-type point-finite precisely-indexed products of ordinals Proposition 2.1.4 pseudo-compact Q-collection Q-cover Q-embedded Q-refinement quasi-uniformity regular cardinal resp result follows St(x stationary subset subspace suffices to show Suppose Suslin tree T₁ Theorem uncountable w)-compact w)-metacompact w)-paracompactness w₁ weakly orthocompact X-open α α αελ ε μ ε ω ελ εω