584 research outputs found
Continuous images of Cantor's ternary set
The Hausdorff-Alexandroff Theorem states that any compact metric space is the
continuous image of Cantor's ternary set . It is well known that there are
compact Hausdorff spaces of cardinality equal to that of that are not
continuous images of Cantor's ternary set. On the other hand, every compact
countably infinite Hausdorff space is a continuous image of . Here we
present a compact countably infinite non-Hausdorff space which is not the
continuous image of Cantor's ternary set
An Easton-like Theorem for Zermelo-Fraenkel Set Theory Without Choice (Preliminary Report)
By Easton's theorem one can force the exponential function on regular
cardinals to take rather arbitrary cardinal values provided monotonicity and
Koenig's lemma are respected. In models without choice we employ a "surjective"
version of the exponential function. We then prove a choiceless Easton's
theorem: one can force the surjective exponential function on all infinite
cardinals to take arbitrary cardinal values, provided monotonicity and Cantor's
theorem are satisfied, irrespective of cofinalities
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