# Factoids #2

2020-12-26, post № 238

mathematics, #bijection, #calculus, #Lipschitz, #naturals

## VII) Cardinality coercion: $\mathbb{R}\hookleftarrow\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R}$

Claim. There exist both $\iota:\{\mathbb{N}\to\mathbb{N}\}\hookrightarrow\mathbb{R}$ together with $\pi:\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R}$.
Proof. Iota. Define $\iota:\{\mathbb{N}\to\mathbb{N}\}\hookrightarrow\mathbb{R}$ via
and observe any sequence’s reconstructibility by dyadic expansion.
Pi. Define $\pi:\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R}$ via
and observe any real’s constructibility by dyadic expansion.

Thus, $\{\mathbb{N}\to\mathbb{N}\}\cong\mathbb{R}$ in $\mathbf{(SET)}$ is shown.

## VIII) Codomain crumpling: $\{\mathbb{N}_0\to\mathbb{N}_0\}\cong\mathrm{Sq}^\star_\Delta(\mathbb{N}_0)$

Let $\mathrm{Sq}^\star_\Delta(\mathbb{N}_0):=\{\mathbb{N}_0\to\mathbb{N}_0\}\cong\{f:\mathbb{N}_0\to\{\star,\Delta\}:\#f^{-1}(\{\Delta\})=\infty\}$.

Claim. There exists $\alpha:\{\mathbb{N}_0\to\mathbb{N}_0\}\ {\sfrac\hookrightarrow\twoheadrightarrow}\ \mathrm{Sq}^\star_\Delta(\mathbb{N}_0)$.
Proof. Define $\alpha:\{\mathbb{N}_0\to\mathbb{N}_0\}\to\mathrm{Sq}^\star_\Delta(\mathbb{N}_0)$ via
and observe any $\mathbb{N}_0$-sequence’s reconstructibility by counting $\Delta$-segmented section’s lengths as well as any $\{\star,\Delta\}$-sequence’s constructibility by defining $\star$-run lengths.

## IX) A non-isomorphic Lipschitzian bijection

Claim. A bijective Lipschitzian map need not be an isomorphism in $\mathbf{(MET)}$ with Lipschitzian arrows, i. e. its inverse may degrade in regularity.
Proof. Let
By $\arctan(\mathbb{R}_{\geq 0})=[0,\sfrac\pi 2[$ and the arctangent’s strict isotonicity together with $\partial\mathrm{B}_1(0)$’s conventional parametrization, $f$’s bijectivity is apparent. Since $f$ is furthermore smooth, $f\mid_{[0,t_1]}$ is globally Lipschitzian for all $t_1\geq 0$. As the local Lipschitz constant approaches zero for $t_1\nearrow\infty$, one can deduce $f$’s global Lipschitz continuity.
Now, looking at a neighborhood $N_{(0,0)}$ of $(0,0)\in\mathbb{R}^2$ one finds for any given $t_1\geq 0$ a radius $\varrho>0$ with its ball $\mathrm{B}_\varrho((0,0))\subset N_{(0,0)}$ satisfying $f^{-1}(\mathrm{B}_\varrho((0,0)))\supset [t_1,\infty[$, proving $f^{-1}$’s non-continuity and thereby its violation of the Lipschitzian property.