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

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
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