Dimension, additive properties of

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

The countable closed sum theorem. If a normal Hausdorff space can be represented as a finite or countable sum of closed subsets , then

If is also perfectly normal or hereditarily paracompact, then

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

If is also perfectly normal and hereditarily paracompact, then

The addition theorem. If the space is Hausdorff, hereditarily normal and if , then

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

A metric space has dimension if and only if

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

Comments

See also Dimension; Dimension theory.

References

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