A blend of methods of recursion theory and topology: A П 0^1 by Kalantari I., Welch L.

By Kalantari I., Welch L.

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Another splitting of the compactification problem of a totally different nature was introduced by Sklyarenko in [1960]. He defined a new invariant which is connected to a characterization of dimension by means of special bases for the open sets of a space. 8. Definition. Let n = -1 or n E N. A separable metrizable space X is said to have SklX 5 n if X has a base B = { U; : i E N} for the open sets such that for any n -t 1 different indices io, , . , i n the intersection B (Ui,,) n . - n B ( U i , ) is compact.

6. Suppose that X is a metrizable space and n is a natural number. Then icd X 5 n if and only if there exists a base B for the open sets of X such that icd B ( U ) 5 n - 1 for every U in B. Our first goal is to resolve the analogue of de Groot’s compactification problem for the completeness degrees. To this end, we define the corresponding deficiency. 10. Definition. With C denoting the class of complete metrizable spaces, the completeness deficiency of a metrizable space X is defined by . C-defX = min{Ind(Y \ X ) :X C Y , Y E C}.

From the observation that B ( U 1 ) = B (Us) it follows that for any k distinct sets out of the collection { B (UO), B (Ul), B (Uz), B (U3)) the intersection has small inductive dimension 2 - k when k 5 3. ) From the base B' we form the base f3 = { Ui : i E N} by adjoining the open set U3 t o it. Then B has the property that any three different boundaries of members of B have empty intersection. Now let us concentrate on B ( U O ) and its base Bo = { U; n B (Uo) : i # 0 }. One would expect BO to be a base for B ( U O )that witnesses the fact that ind B ( U O )= 1 when one uses Odim.