#paper #tree #souslin #aronszajn # Abstract Tree spaces provide a useful collection of 'standard examples' of topological spaces with specific properties: a fact which was already noticed by F. B. Jones in the 1930's. Certain properties of tree spaces depend upon the structural property of the tree concerned (as a partially ordered set). We obtain characterizations of those trees whose topologies are (a) normal, (b) collectionwise Hausdorff topologies. (Both results relate closely to the normal Moore space problem, and arose out of our research on that problem, the results of which appear elsewhere.) The structural properties of trees involved in these results are generalizations of the Souslin condition on trees. 2.1. THEOREM. Let $T$ be an $\omega_{1}$ tree. $T$ is a Souslin tree if and only if whenever $A,B$ are disjoint closed subsets of the space $T$, $\hat{A} \cap \hat{B}$ is countable (where $\hat{A}=\bigcap_{x\in A} \hat{x}$ and $\hat{x}$ is the downward closure of $x$). 2.2. COROLLARY. If $T$ is a Souslin tree, its topology is normal. 3.1. LEMMA. if there exists an almost Souslin tree, then there exists an almost Souslin tree which is not a Souslin tree. ![[Pasted image 20230413162413.png]] 3.2. LEMMA. If T is a special Aronszajn tree, then T is not an almost Souslin tree. 3.3. THEOREM. Let T be an $\omega_{1}$ tree. T is an almost Souslin tree if and only if its tree topology is a collectionwise Hausdorff topology.