Ordinals and scattered compact ordered spaces
The author studies scattered compact linearly ordered spaces (SCOS). A scattered space is one which has no dense-in-itself subset. He relates these spaces to compact well-ordered spaces and to spaces of ordinals. His first theorem states that a SCOS is homeomorphic to a compact well- ordered space only if it has no pit-points. A pit-point has infinite left and right cofinalities, one of which is uncountable. The characteristic ch(K) of a space K is \(<\mu,m>\) if \(\mu\) is the first ordinal such that the derived set \(K^{(\mu +1)}\) is empty, while \(K^{(\mu)}\) has m elements. If ch(K) of a SCOS space K is \(<\mu,m>\), the author shows that there is a continuous mapping of K onto \(\omega^{\mu}\cdot m+1\). If s is the number of symmetric pit-points of K, there is a continuous mapping of \(\omega^{\mu}\cdot(m+s)+1\) onto K, and K contains a closed subspace W homeomorphic to \(\omega^{\mu}\cdot m+1\). He states without proof that K is a SCOS only if K is an order-two continuous image of a compact well- ordered space.
scattered compact linearly ordered spaces
scientific article; zbMATH DE number 3868204
scientific article