Local dimension

of a normal topological space

The topological invariant , defined as follows. One says that , if for any point there is a neighbourhood for which the Lebesgue dimension of its closure satisfies the relation . If for some , then the local dimension of is finite, so one writes and puts

Always ; there are normal spaces with ; in the class of paracompact spaces always . If in the definition of local dimension the Lebesgue dimension is replaced by the large inductive dimension , then one obtains the definition of the local large inductive dimension .

Comments

See [a1] for a construction of a space with and — as an application — a hereditarily normal space with yet contains subspaces of arbitrary high dimension.

For the notions of the local dimension at a point of an analytic space, algebraic variety or scheme cf. Analytic space; Dimension of an associative ring; Analytic set, and Spectrum of a ring.

References

[a1]	E. Pol, R. Pol, "A hereditarily normal strongly zero-dimensional space containing subspaces of arbitrarily large dimension" Fund. Math. , 102 (1979) pp. 137–142
[a2]	R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50