On Two Problems of Urysohn

Abstract

Urysohn [4, p. 290] has considered the problem of determining the most general class of topological spaces in which non-constant real-valued continuous functions exist. He showed that a countable Hausdorff space exists in which every continuous real-valued function is constant [4, pp. 274-283]. Pospisil [2, pp. 1-3] has extended this result by exhibiting a Hausdorff space of arbitrary infinite cardinal number in which every continuous real-valued function is constant. The spaces constructed by both Urysohn and Posp ?il, however, fail to satisfy the Urysohn separation axiom.2 Urysohn posed, but left unanswered, the question of whether or not regular topological spaces exist in which every continuous real-valued function is constant. In the present paper, it will be proved that regular spaces exist in great profusion on which every real-valued continuous function is constant. Urysohn also considered the problem of determining the least cardinal number of a connected space. [4, pp. 274-283], and, by the construction mentioned above, showed that a countable connected Hausdorff space exists. This example, however, does not satisfy the Urysohn separation axiom. Urysohn's result is extended in the present paper by the construction of a countable -Urysohn space in which every continuous function is constant and which is accordingly connected. THEORBM 1. Let N be an infinite cardinal number which is not the sum of No cardinal numbers smaller than R. Then there exists a regular space 'k of cardinal number N in which every continuous real-valued function is constant. Our construction of the space ?k, which proceeds in several steps, makes essential use of a space constructed by Tychonoff [3, pp. 553-556] and the condensation process devised by Urysohn [4, pp. 274-283]. First, let A be the least ordinal number with corresponding cardinal number N. It is obvious that A is not the limit of a countable sequence of ordinal numbers smaller than A. Let D be the set of all ordinal numbers a such that 1 ? a < A Let C be the set of all ordinal numbers a such that 1 < a ? c, where X denotes, as usual, the least infinite ordinal number. D and C are made into topological spaces by the following construction. In D, every point 6, where 1 < a < A, is isolated. Neighborhoods Uy(A) of the point A e D are the sets E[b; y < _5 ?A]. C is given a similar neighborhood system with X as the only non-isolated point. Let S be the set of all pairs (6,a) of ordinal numbers such that ( E D and a E C.