Two new equivalents of Lindelöf metric spaces |
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| Abstract: | In the realm of Lindelöf metric spaces the following results are obtained in  : (i) If  is a Lindelöf metric space then it is both densely Lindelöf and almost Lindelöf. In addition, under the countable axiom of choice  , the three notions coincide. (ii) The statement “every separable metric space is almost Lindelöf” implies that every infinite subset of  has a countably infinite subset). (iii) The statement “every almost Lindelöf metric space  is quasi totally bounded implies  . (iv) The proposition “every quasi totally bounded metric space is separable” lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem  and  . Likewise, the statement “every pre‐Lindelöf (or Lindelöf) metric space is separable” lies strictly between  and  . |
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