By Ross Honsberger

Ross Honsberger has performed it back. He has introduced jointly one other significant number of ordinary mathematical difficulties and their suggestions abounding in awesome surprises and incredible rules that mirror the great thing about arithmetic. a lot of those difficulties come from mathematical journals. Others come from a number of mathematical competitions akin to the match of the cities, the Balkan Olympiad, the yank Invitational arithmetic examination, and the Putnam examination. And, after all, there's a challenge steered by means of Paul Erd?s. This booklet is perfect for college students, lecturers and a person drawn to leisure arithmetic.

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**Example text**

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.