Local dimension

2020 Mathematics Subject Classification: Primary: 54F45 [MSN][ZBL]

of a normal topological space $X$

The topological invariant $\mathrm{locdim}(X)$, defined as follows. One says that $\mathrm{locdim}(X) \le n$, $n = -1,0,1,\ldots$ if for any point $x \in X$ there is a neighbourhood $O_x$ for which the Lebesgue dimension of its closure satisfies the relation $\dim \bar O_x \le n$. If $\mathrm{locdim}(X) \le n$ for some $n$, then the local dimension of $X$ is finite, so one writes $\mathrm{locdim}(X) < \infty$ and puts $$ \mathrm{locdim}(X) = \min\\{ n : \mathrm{locdim}(X) \le n \\} $$

Always $\mathrm{locdim}(X) \le \dim(X)$; there are normal spaces $X$ with $\mathrm{locdim}(X) < \dim(X)$; in the class of paracompact spaces always $\mathrm{locdim}(X) = \dim(X)$. If in the definition of local dimension the Lebesgue dimension $\dim \bar O_x $ is replaced by the large inductive dimension $\mathrm{Ind} \bar O_x $, then one obtains the definition of the local large inductive dimension $\mathrm{locInd}(X)$.

#### Comments[edit]

See [a1] for a construction of a space with $\mathrm{locdim}(X) < \dim(X)$ and — as an application — a hereditarily normal space $Y$ with $\dim Y = 0$ yet $Y$ 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[edit]

[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

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[a2] |  R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50