Finite-valued mappings preserving dimension. (English) Zbl 1213.54049

Spěvák, Jan

Houston J. Math. 37, No. 1, 327-348 (2011).

The author defines a domination relation between Tychonoff spaces: \(X\) dominates \(Y\) if there are finite-valued maps \(F:X\Rightarrow Y\) and \(G:Y\Rightarrow X\) such that for every \(y\in Y\) there is an \(x\in X\) such that \(x\in G(y)\) and \(y\in F(x)\) and both \(X\) and \(Y\) have countable covers by cozero-sets such that the restrictions.of \(F\) and \(G\) to members of these covers are lower semi-continuous. He proves that \(\dim X\geq\dim Y\) whenever \(X\) dominates \(Y\) and hence \(\dim X=\dim Y\) if the two spaces dominate each other.

Reviewer: K. P. Hart (Delft)

MSC:

54F45	Dimension theory in general topology
54C10	Special maps on topological spaces (open, closed, perfect, etc.)
54C60	Set-valued maps in general topology

Keywords:

covering dimension; lsc mapping