计算超重空间

摘要：

令c= 2 ^aleph 0表示连续统的基数，令a，b，k是无限基数，a<b \leq 2 ^a。我们证明了在同胚下存在大小为a、权为b的2 ^b个T0-空间。在这些非同胚空间中，我们找到了（1）2^b零维、散布、仿紧、完全正规空间（2）2 ^b连通和局部连通的T2空间;（3）2 ^b路径连通和局部路径连通的仿紧完美正规空间，只要a \geq c;（4）2 ^b连通、处处局部连通、全路径不连通、仿紧、完美正规空间，条件是a \geq c;（5）2 ^b离散、紧T1-空间;（6）2 ^b连通、局部连通、紧T1-空间;（7）2 ^b路径连通、离散、紧T0-空间;（8）当a^{<k}=a且b^{<k}=b且2^b>2^a时，设P_k-空间为2^b-散布仿紧空间.

英文摘要：

Let c=2^aleph0 denote the cardinality of the continuum and let a,b,k be infinite cardinal numbers with a<b\leq 2^a. We show that there exist precisely 2^b T0-spaces of size a and weight b up to homeomorphism. Among these non-homeomorphic spaces we track down (1) 2^b zero-dimensional, scattered, paracompact, perfectly normal spaces (which are also extremally disconnected in case that b=2^a); (2) 2^b connected and locally connected Hausdorff spaces; (3) 2^b pathwise connected and locally pathwise connected, paracompact, perfectly normal spaces provided that a\geq c; (4) 2^b connected, nowhere locally connected, totally pathwise disconnected, paracompact, perfectly normal spaces provided that a\geq c; (5) 2^b scattered, compact T1-spaces; (6) 2^b connected, locally connected, compact T1-spaces; (7) 2^b pathwise connected and scattered, compact T0-spaces; (8) 2^b scattered, paracompact P_k-spaces whenever a^{<k}=a and b^{<k}=b and 2^b>2^a.

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