Dimension, additive properties of

Properties expressing a connection between the dimension of a topological space $ X $ that can be represented as the sum of subspaces $ X _ \alpha $ with the dimensions of these subspaces $ X _ \alpha $. There are several types of additive properties of dimension.

The countable closed sum theorem. If a normal Hausdorff space $ X $ can be represented as a finite or countable sum of closed subsets $ X _ {i} $, then

$$ \mathop{\rm dim} X = \sup _ { i } \mathop{\rm dim} X _ {i} . $$

If $ X $ is also perfectly normal or hereditarily paracompact, then

$$ \mathop{\rm Ind} X = \sup _ { i } \mathop{\rm Ind} X _ {i} . $$

The locally finite closed sum theorem. If a normal Hausdorff space $ X $ can be represented as the sum of a locally finite system of closed subsets $ X _ \alpha $, then

$$ \mathop{\rm dim} X = \sup _ \alpha \mathop{\rm dim} X _ \alpha . $$

If $ X $ is also perfectly normal and hereditarily paracompact, then

$$ \mathop{\rm Ind} X = \sup _ \alpha \mathop{\rm Ind} X _ \alpha . $$

The addition theorem. If the space $ X $ is Hausdorff, hereditarily normal and if $ X = A \cup B $, then

$$ \mathop{\rm dim} X \leq \mathop{\rm dim} A + \mathop{\rm dim} B + 1 $$

(the Menger–Urysohn formula). If $ X $ is also perfectly normal, then

$$ \mathop{\rm Ind} X \leq \mathop{\rm Ind} A + \mathop{\rm Ind} B + 1. $$

A metric space $ R $ has dimension $ \mathop{\rm dim} R \leq n $ if and only if

$$ R = \cup _ {i = 1 } ^ { {n } + 1 } R _ {i} ,\ \ \mathop{\rm dim} R _ {i} \leq 0,\ \ i = 1 \dots n + 1; \ n = 0, 1 , . . . . $$

If $ X $ is hereditarily normal and Hausdorff, then for any closed subset $ F $ one has

$$ \mathop{\rm dim} X = \max ( \mathop{\rm dim} F, \mathop{\rm dim} X \setminus F ), $$

$$ \mathop{\rm Ind} X = \max ( \mathop{\rm Ind} F, \mathop{\rm Ind} X \setminus F ). $$

Comments

See also Dimension; Dimension theory.

References

[a1]	R. Engelking, "Dimension theory" , North-Holland & PWN (1978)