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It follows from the sum theorem that the subspace Y = U{Fr U: U E B ] has dimension < n - 1 and from \ has dimension < 0. 2. 8. The second decomposition theorem. A separable metric space X satisfies the inequality indX < n > 0 if and only if X can be represented 44 Dimension theory of separable metric spaces as the union of n + 1 subspaces Z , , Z,, ... , n + l . 0 [Ch. 20), the Hilbert cube cannot be represented as a countable union of zero-dimensional subspaces, or-equivalently-of finite-dimensional subspaces.

For every point x in Euclidean n-space R" or in the n-sphere S" and each neighbourhood V of the point x there exists a neighbourhood U of x such that U c V and the boundary FrU is homeomorphic to S"-'. Hence, as shown by an inductive argument, indR" < n, indS" < n and indP < n for every natural number n. 8. The equality indR" = n is of utmost importance for dimension theory. In a sense, it justifies the definition of the dimension function by showing that this definition yields a notion conforming to geometric intuition.

Definition. A topological space X is called hereditarily disconnected if X does not contain any connected subspace of cardinality larger than one. Every totally disconnected space is hereditarily disconnected. Indeed, if X is a totally disconnected space, then for each subspace M c X which contains at least two distinct points x, y the sets M n U and M\U, where U is an open-and-closed subset of X such that x E U and y EX\ U, form a decomposition of the space M into two non-empty disjoint open subsets, so that the space M is not connected.

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