Jonathan. Frech’s WebBlog

Lower semicontinuity’s misremembrance (#289)

Jonathan Frech,

Snails are surprisingly fast: you look at them and they creep so slowly as to strain your attention. Yet you look away for an undefined moment and their progress in distance appears wholly be­wil­der­ing. Then again, I would wager to outrun most snails in a 200-meter dash.

I think symmetry’s ubiquity is an equally fickle observation; you may on­ly be able to bear wit­ness to it once you have already manifested it for yourself. Notably notationally; among the following lines

$$\liminf_{y\to x\neq y}f(y)\leq f(x) \quad\textbf{(Q)}$$$$\liminf_{y\to x\neq y}f(y)\leq f(x) \quad\textbf{(Q)}$$
$$f(x)\leq\liminf_{y\to x\neq y}f(y) \quad\textbf{(C)}$$$$f(x)\leq\liminf_{y\to x\neq y}f(y) \quad\textbf{(C)}$$

for a map $f:X\to\mathbb{R}$$f:X\to\mathbb{R}$ from a topological space $X$$X$ to the real line and a fixed point $x\in X$$x\in X$, precisely one is the definition of lower semicontinuity, the oth­er but a figment of a symmetry-ravenous mind.

I knew for a long time about $\textbf{(Q)}$$\textbf{(Q)}$’s questionability, but on­ly two days ago I realized how separate a statement it is. On­ly the obvious implication $\lnot\textbf{(C)}\implies\textbf{(Q)}$$\lnot\textbf{(C)}\implies\textbf{(Q)}$ holds with all three oth­er conjunctions being possible: $\textbf{(C)}\land\lnot\textbf{(Q)}$$\textbf{(C)}\land\lnot\textbf{(Q)}$ satisfied by $1-\chi_{\{0\}}$$1-\chi_{\{0\}}$ on $X=\mathbb{R}$$X=\mathbb{R}$ at $x=0$$x=0$, $\textbf{(C)}\land\textbf{(Q)}$$\textbf{(C)}\land\textbf{(Q)}$ satisfied by $\chi_{]{}0,\infty{}[}$$\chi_{]{}0,\infty{}[}$ on $X=\mathbb{R}$$X=\mathbb{R}$ at $x=0$$x=0$ and $\lnot\textbf{(C)}\land\textbf{(Q)}$$\lnot\textbf{(C)}\land\textbf{(Q)}$ satisfied by $\chi_{[0,\infty{}[}$$\chi_{[0,\infty{}[}$ on $X=\mathbb{R}$$X=\mathbb{R}$ at $x=0$$x=0$.

Curiously, the infamous $\chi_{\mathbb{Q}}$$\chi_{\mathbb{Q}}$ satisifies $\textbf{(Q)}$$\textbf{(Q)}$ at every point, intuitively showing $\textbf{(Q)}$$\textbf{(Q)}$’s in­ap­ti­tude of representing an (even weakened) in­ter­pre­ta­tion of continuity.