Factoids #3 (#246)
Jonathan Frech
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.
“⇐”. Define
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.
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.