2021-07-10, post № 246
mathematics, #topology, #reals, #lipschitzian
X) An intriguingly delicate arbitrarily small countable open cover of uncountably many unbounded points inferred from locally lipschitzian maps’ inability to increase a set’s Hausdorff dimension
Claim. For any uncountable Lebesgue measure zero set , there exists a point such that is still uncountably infinite. Define and on this right portion as . Looking at the graph , it is a Hausdorff measure zero set — meaning the entirety of all uncountably many points clustered at span an infinite vertical distance and yet are coverable by countably many sets of arbitrarily small diameter in sum.
Proof. Since is locally lipschitzian and the reals are Lebesgue -compact, its image cannot increase the one-dimensional Hausdorff measure which is presumed to vanish.
XI) A closed set in the plane whose projection onto the first coordinate is open in the reals
Claim. There exists a set which is closed whilst its projection under is open, more precisely .
Proof. Define and note that this union of closed balls with pairwise positive distance is again closed via sequence closedness. Furthermore, is apparent and for any there exists an index such that and thus since it lies in the projection of a closed ball horizontally centered at a half with radius .
XII) Characterizing separability by subsequence approximabilty
Claim. A general topological space is separable iff there exists a sequence such that for any there exists a subsequence which satisfies .
Proof. “⇒”. Let be countable and dense. Let be a (global) enumeration, its evaluation denoted by . Since lies densely, holds. For arbitrary point and open neighborhood follows the existence of a and thus for a subsequence.“⇐”. Define . This set is by definition countable and for arbitrary there exists which approximates . Following, for an open neighborhood an index exists with , showing and thereby that lies dense.
XIII) Discontinous inclusion
Claim. On a set , endow two nontrivially progressively fine topologies . Embedding is a discontinuous undertaking.
Proof. For arbitrary open set , it follows , thus is discontinuous.
Factoid XIII) was originally formulated June 2020.
Claim. Any non-discrete topology endowed on a set can be embedded into a finer topology on the same set such that this inclusion is discontinuous.
Proof. From , one gets the existence of a point whose singleton is not open. Refining where denotes the generated topology together with the previously shown yields the claim.