Concerning the Separability of Certain Locally Connected Metric Spaces
نویسندگان
چکیده
If a connected metric space S is locally separable, then S is separable. If a connected, locally connected, metric space S is locally peripherally separable, then S is separable. Furthermore if a connected, locally connected, complete metric space S satisfies certain "flatness" conditions, it is known to be separable. These "flatness" conditions are rather strong and involve both im kleinen and im grossen properties, which makes application rather awkward in some cases. If, however, this space S contains no skew curve of type 1, then S has a certain amount of "flatness," but not quite enough to imply separability as can be seen from the following example. Let S consist of the points of the 2-sphere, distance being redefined as follows: (1) if the points X and Y of 5 lie on the same great circle through the poles, then d(X, Y) is the ordinary distance on the sphere but (2) if the points lie on different great circles through the poles, then d(X, Y) is the sum of the ordinary distances from each point to the same pole, using the pole which gives the smaller sum. The space 5 is a connected, locally connected, complete metric space which contains no skew curve of type 1 but 5 is not separable. Furthermore, S contains no cut point. However, if this last condition is strengthened slightly, separability follows as is seen in the following theorem.
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