ONT Re: Topology
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| 1. Topological Spaces
|
| 1.6. Bases and Subbases (cont.)
|
| A space whose topology has a countable base has
| many pleasant properties. Such spaces are said
| to satisfy the 'second axiom of countability'.
| (The terms 'separable' and 'perfectly separable'
| are also used in this connection, but we shall
| use neither.)
|
| 13. Theorem. If A is an uncountable subset of a space whose topology has
| a countable base, then some point of A is an accumulation point of A.
|
| Proof. Suppose that no point of A is an accumulation point and that !B! is
| a countable base. For each x in A there is an open set containing no point
| of A other than x, and since !B! is a base we may choose B_x in !B! such that
| B_x |^| A = {x}. There is then a one-to-one correspondence between the points
| of A and the members of a subfamily of !B!, and A is therefore countable. þ
|
| A sharper form of this theorem is stated in problem 1.H.
|
| JLK, Gen Top, pages 48-49.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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