Two pathological countable topological spaces are constructed. Each is quasimetrizable and has a simple explicit quasimetric. One is a locally connected Hausdorff space and is an extension of the rationals. The other is a connected space which becomes totally disconnected upon the removal of a single point. This space satisfies the Urysohn separation property-a property between Tí and T¡-and is an extension of the space of rational points in the plane. Both are one dimensional in the Menger-Urysohn [inductive] sense and infinite dimensional in the Lebesgue [covering] sense.

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Two pathological countable topological spaces are constructed. Each is quasimetrizable and has a simple explicit quasimetric. One is a locally connected Hausdorff space and is an extension of the rationals. The other is a connected space which becomes totally disconnected upon the removal of a single point. This space satisfies the Urysohn separation property-a property between Tí and T¡-and is an extension of the space of rational points in the plane. Both are one dimensional in the Menger-Urysohn [inductive] sense and infinite dimensional in the Lebesgue [covering] sense.

Communicated by B. MondIn this note we define a rather pathological connectedness property of Hausdorff spaces which is stronger than ordinary connectedness. We obtain a few basic properties of such spaces and derive a method for constructing them. It turns out that countable Hausdorff spaces having the connectedness property are as easily constructed as uncountable ones. Hence we have still another method for constructing countable connected Hausdorff spaces.DEFINITION. The subset K of the space X saturates X if K is infinite and for each open set U in X, K-U is finite. The space X is said to be saturated if X is a point or some subset of X saturates X. THEOREM 1. If U is an open subset of the saturated space X, then U is connected.PROOF. Suppose K is a subset of X that saturates X and C/is an open subset of X such that U is not connected. Then U is the union of two disjoint closed sets R and S. U-S is open, U-S £ R, and hence K-R is finite. Similarly, K-S is finite. But K = (K-R) u {K-S).COROLLARY. Every saturated space is connected. THEOREM If x and y are two points of the saturated space X and £f is an open cover of X, then there is a simple chain of elements of Sf from x to y with at most three links.PROOF. Suppose x e Ue Sf andy e We £f. Since l / u (fis connected, there is a point z such that zeU nW. Let V be an element of Sf containing z, and form the simple chain from the sets U, V, and W.