# Algebraische Topologie by Wolfgang Lück

By Wolfgang Lück

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Top. , 4, 315–339. [1974b] On the existence of non-metrizable hereditarily Lindel¨of spaces with point-countable bases. Duke Math. , 41, 299–304. [1974c] On the existence of normal metacompact Moore spaces which are not metrizable. Can. J. , 26, 1–6. [1976a] The density topology. Pac. J. , 62, 175–184. [1976b] Stalking the Souslin tree - a topological guide. Canad. Math. , 19, 337–341. [1976c] Weakly collectionwise Hausdorﬀ spaces. Top. , 1, 295–304. [1977a] First countable spaces with calibre ℵ1 may or may not be separable.

Rst countable spaces are ℵ1 -collectionwise Hausdorﬀ ? This question is just something that I expected would have a positive answer but couldn’t make any headway on. Shelah [1979] showed that ♦∗ implies that normal ﬁrst countable spaces are ℵ1 -collectionwise Hausdorﬀ and every other separation theorem which used normality eventually was extended to countable paracompactness (see Watson [1985] and Burke’s use of PMEA). The real question here is vague: “is there a distinction between the separation properties of normality and countable paracompactness”.

One expects ﬁrst countability to be a big help but so far it seems useless in this context. The drawback to this question is that if the answer is no, one ﬁrst has to solve the set-theoretic question and then ﬁgure out how to lower the character from the continuum to ℵ0 . Getting the character down is always interesting. On the other hand, if there is a theorem, that might involve a hard look at the weak version of ♦ invented by Keith Devlin (Devlin and Shelah [1978]) and lots of people would be interested in an essential use of that axiom.