Graphical convergence

The space of continuous functions (say, on {[0,1]}) is usually given the uniform metric: {d_u(f,g) = \sup_{x}|f(x)-g(x)|}. In other words, this is the smallest number {\rho} such that from every point of the graph of one function we can jump to the graph of another function by moving at distance {\le \rho} in vertical direction.

Uniform metric: vertical distances
Uniform metric is based on vertical distances

Now that I put it this way, why don’t we drop “in vertical direction”? It’ll still be a metric, namely the Hausdorff metric between the graphs of {f} and {g}. It’s natural to call it the graphical metric, denoted {d_g}; from the definition it’s clear that {d_g\le d_u}.

Graphical metric: Hausdorff distance
Weird metric, weird color

Some interesting things happen when the space of continuous functions is equipped with {d_g}. For one thing, it’s no longer a complete space: the sequence {f_n(x)=x^n} is Cauchy in {d_g} but has no limit.

Sequence of x^n
Sequence of x^n

On the other hand, the bounded subsets of {(C[0,1],d_g) } are totally bounded. Indeed, given {M>0} and {\epsilon>0} we can cover the rectangle {[0,1]\times [-M,M]} with a rectangular mesh of diameter at most {\epsilon}. For each function with {\sup|f|\le M}, consider the set of rectangles that its graph visits. There are finitely many possibilities for the sets of visited rectangles. And two functions that share the same set of visited rectangles are at graphical distance at most {\epsilon} from each other.

Boxy approximation to the graph
Boxy approximation to the graph

Thus, the completion of {C[0,1]} in the graphical metric should be a nice space: bounded closed subsets will be compact in it. What is this completion, concretely?

Here is a partial answer: if {(f_n)} is a graphically Cauchy sequence, its limit is the compact set {\{(x,y): g(x)\le y\le h(x)\}} where

\displaystyle    g(x) = \inf_{x_n\rightarrow x} \liminf f_n(x_n)

(the infimum taken over all sequences converging to {x}), and

\displaystyle    h(x) = \sup_{x_n\rightarrow x} \limsup f_n(x_n)

It’s not hard to see that {g} is upper semicontinuous and {h} is lower semicontinuous. Of course, {g\le h}. It seems that the set of such pairs {(g,h)} indeed describes the graphical completion of continuous functions.

For example, the limit of {f_n(x)=x^n} is described by the pair {g(x)\equiv 0}, {h(x)=\chi_{\{1\}}}. Geometrically, it’s a broken line with horizontal and vertical segments

For another example, the limit of {f_n(x)=\sin^2 nx} is described by the pair {g(x)\equiv 0}, {h(x)\equiv 1}. Geometrically, it’s a square.

Stripey sines
Stripey sines

Strength in unity

Elementary point-set topology today, on a metric space X.

  • X is compact if every open cover of X has a finite subcover. One could just as well say “every sequence has a convergent subsequence”, but it’s considered bad form.
  • X is totally bounded if for every r>0 the space can be covered by finitely many balls of radius r. One could just as well say “every sequence has a Cauchy subsequence”, but it’s probably bad form, too.
  • X is complete if every Cauchy sequence converges. I don’t know if anyone considers this sequential definition bad form.

A look at the sequential forms of definitions reveals that a space is compact iff it is complete and totally bounded.

Which of the above properties survive under continuous maps?

  • If X is compact, then every continuous image of X is also compact.
  • If X is totally bounded, its continuous image may fail to be totally bounded, or even bounded. For example, the interval (0,1] is continuously mapped onto [1,\infty) by x\mapsto x^{-1}.
  • if X is complete, its continuous image may fail to be complete. For example, the interval [1,\infty) is continuously mapped onto (0,1] by x\mapsto x^{-1}.

There must be other natural examples of how two properties P_1 and P_2 are not invariant (under some class of transformations) on their own, but are invariant together. None come to mind at this moment.