Szlenk and w^⁎-dentability indices of C(K)

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• 作者：R.M. Causey ^{causeyrm@miamioh.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Szlenk index ; Dentability index
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：834-845
• 全文大小：375 K

文摘

Given any compact, Hausdorff space m>K  m> and n id="mmlsi1" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306278&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306278&_rdoc=1&_issn=0022247X&md5=025423f6ec5c4353f8a17e02fda5a15c" title="Click to view the MathML source">1<p<&infin;n>n class="mathContainer hidden">n class="mathCode"><math altimg="si1.gif" overflow="scroll"><mn>1mn><mo><mo><mi>pmi><mo><mo><mo>&infin;mo>math>n>n>n>, we compute the Szlenk and n id="mmlsi2" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306278&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306278&_rdoc=1&_issn=0022247X&md5=f3ab4836e0a14df4ab4290550ad15f22" title="Click to view the MathML source">w^⁎n>n class="mathContainer hidden">n class="mathCode"><math altimg="si2.gif" overflow="scroll"><msup><mrow><mi>wmi>mrow><mrow><mo>⁎mo>mrow>msup>math>n>n>n>-dentability indices of the spaces n id="mmlsi3" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306278&_mathId=si3.gif&_user=111111111&_pii=S0022247X16306278&_rdoc=1&_issn=0022247X&md5=6e40aa03ed14c811aaec00298964abc5" title="Click to view the MathML source">C(K)n>n class="mathContainer hidden">n class="mathCode"><math altimg="si3.gif" overflow="scroll"><mi>Cmi><mo stretchy="false">(mo><mi>Kmi><mo stretchy="false">)mo>math>n>n>n> and n id="mmlsi4" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306278&_mathId=si4.gif&_user=111111111&_pii=S0022247X16306278&_rdoc=1&_issn=0022247X&md5=5eb4e1233390f0c225fab5bced1b822f" title="Click to view the MathML source">L_p(C(K))n>n class="mathContainer hidden">n class="mathCode"><math altimg="si4.gif" overflow="scroll"><msub><mrow><mi>Lmi>mrow><mrow><mi>pmi>mrow>msub><mo stretchy="false">(mo><mi>Cmi><mo stretchy="false">(mo><mi>Kmi><mo stretchy="false">)mo><mo stretchy="false">)mo>math>n>n>n>. We show that if m>K  m> is compact, Hausdorff, scattered, n id="mmlsi5" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306278&_mathId=si5.gif&_user=111111111&_pii=S0022247X16306278&_rdoc=1&_issn=0022247X&md5=79c60df45187652e54da58783deac1d5" title="Click to view the MathML source">CB(K)n>n class="mathContainer hidden">n class="mathCode"><math altimg="si5.gif" overflow="scroll"><mi>Cmi><mi>Bmi><mo stretchy="false">(mo><mi>Kmi><mo stretchy="false">)mo>math>n>n>n> is the Cantor&ndash;Bendixson index of m>Km>, and m>ξ  m> is the minimum ordinal such that n id="mmlsi342" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306278&_mathId=si342.gif&_user=111111111&_pii=S0022247X16306278&_rdoc=1&_issn=0022247X&md5=dc9089f33ef2453c6e383c9aa2228ea3" title="Click to view the MathML source">CB(K)⩽ω^ξn>n class="mathContainer hidden">n class="mathCode"><math altimg="si342.gif" overflow="scroll"><mi>Cmi><mi>Bmi><mo stretchy="false">(mo><mi>Kmi><mo stretchy="false">)mo><mo>⩽mo><msup><mrow><mi>ωmi>mrow><mrow><mi>ξmi>mrow>msup>math>n>n>n>, then n id="mmlsi7" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306278&_mathId=si7.gif&_user=111111111&_pii=S0022247X16306278&_rdoc=1&_issn=0022247X&md5=38c39d7f80a4b5f8c0c2621348a7b6f3" title="Click to view the MathML source">Sz(C(K))=ω^ξn>n class="mathContainer hidden">n class="mathCode"><math altimg="si7.gif" overflow="scroll"><mi>Smi><mi>zmi><mo stretchy="false">(mo><mi>Cmi><mo stretchy="false">(mo><mi>Kmi><mo stretchy="false">)mo><mo stretchy="false">)mo><mo>=mo><msup><mrow><mi>ωmi>mrow><mrow><mi>ξmi>mrow>msup>math>n>n>n> and n id="mmlsi8" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306278&_mathId=si8.gif&_user=111111111&_pii=S0022247X16306278&_rdoc=1&_issn=0022247X&md5=2d8030d0aaee2d9e3b0088d90e7e6c84" title="Click to view the MathML source">Dz(C(K))=Sz(L_p(C(K)))=ω^1+ξn>n class="mathContainer hidden">n class="mathCode"><math altimg="si8.gif" overflow="scroll"><mi>Dmi><mi>zmi><mo stretchy="false">(mo><mi>Cmi><mo stretchy="false">(mo><mi>Kmi><mo stretchy="false">)mo><mo stretchy="false">)mo><mo>=mo><mi>Smi><mi>zmi><mo stretchy="false">(mo><msub><mrow><mi>Lmi>mrow><mrow><mi>pmi>mrow>msub><mo stretchy="false">(mo><mi>Cmi><mo stretchy="false">(mo><mi>Kmi><mo stretchy="false">)mo><mo stretchy="false">)mo><mo stretchy="false">)mo><mo>=mo><msup><mrow><mi>ωmi>mrow><mrow><mn>1mn><mo>+mo><mi>ξmi>mrow>msup>math>n>n>n>.