Compactness

Management Crisis in Reals

Let me offer you an ice-cream cup. Before handing it over, let me steal away; not part of the ice-cream, just one point from the bottom of the container-cup. Interesting now? You have ice-cream to be taken from the cup as well as from your hand, your shirt, and $\ldots$. The next day, allow me to take away just one point from one of the tubes of your car or bike. Remember, only one point taken away! Have you experienced the collapse of the castle of cards, on removable of a single card? From a bee-hive, pluck out one bee and keep calm!! Let us remove just one point $0$ from $[0, 1]$; and study the crisis caused. The turmoil suggests that before displacing human beings for development purpose, the government officers need take a course on `neighbourhoods'.


We may recall our discussion, that no-beginning property of the set $(0, 1]$, provides it a wild character there. And conversely, the set $[0, 1]$, although containing the entire infinite and wild set $(0, 1)$, and $\{0\}$ and $\{1\}$ added to it does no longer remain wild. Addition of extra points, to an already wild situation does not always make the situation more explosive; rather it tames the wildness.

Let us further probe, whether addition of any other point say $-6$ to the set $(0, 1]$ makes it well-behaved. Obviously not. However, if we define the neighbourhood of $- 6$ as sets of the form $\{- 6 \} \cup (0, p)$, $0 < p \leq 1$; it is seen to work. It clearly demonstrates that the wildness referred here is neighbourhood-related or topological in character. Again, we may observe that the removal of an entire interval $(1/3, 2/3)$ from $[0, 1]$ or the removal of $-6$ from $\{-6\} \cup [0, 1]$ does not affect the nicety or well-manageability
of the set.

We may observe how the non-beginningness in the set provides for wildness by considering the function $f(x) = 1/x $ for $x$ in $(0, 1]$.

Wildness of $f$ near $0$.

The function goes wild as values of the function are considered on points nearer to zero. It allows the continuous function to become unbounded. What needs to be observed is that greater and greater values for the function have become possible due to no-beginningness or no first pointedness of the set $(0, 1]$.

If this non-beginningness is blunted by the inclusion of the point $0$ in the set, and whatever value, say $1000$, is assigned to the function at $0$, inspite of all the freedom to the continuous function to take arbitrary large values and make oscillations, the graph is forced to turn to end up and bang its heads at $1000$, making the function bounded. .

The tamed behaviour of a function on the closed interval $[0, 1]$.

This is how taming works by inclusion of the point $0$: forces the declaration of a value at $0$, and forces the function to ultimately return to this value. I am reminded of a wise father, who used his ingenuity to discipline the wild behaviour of his son. The son was granted all the freedom but was required to return to an arbitrary but fixed declared place of his choice at an arbitrary but to be declared fixed time in each $24$ hours. The reporting at a fixed place, on a fixed time was a must. As the fixed time was approaching, the forced movement to the assigned place had to be started. The managers at reform houses may take the hint.

The niceties of $[0, 1]$ are obviously true for finite sets. Functions defined on finite sets are trivially bounded. Easy to manage sets of the form $[0, 1]$ called compact sets are generalised form of a finite set in the sense that it requires only a finite number of non-trivial sets (open intervals) to contain the set $[0, 1]$. The finiteness condition is trivially fulfilled if open intervals chosen are large in size. So while requiring the finiteness of the covering family, we must insist that only finite family is required even if we choose

1. to cover by smaller and smaller sized open intervals;
2. to cover the set in an arbitrary manner.

The requirement $(1)$ takes care of the wildness of size. It is easily seen that $(-\infty, \infty)$, $\mathbb Z$ or $\mathbb N$ do not satisfy $(1)$; while the open interval $(0, 1)$ or the closed interval $[0, 1]$ both satisfy $(1)$.

$\mathbb{N}$ does not satisfy $(1)$.

Intervals $(0, 1)$ and $[0, 1]$ satisfy $(i)$ for arbitrary $p > 0$.

The requirement $(2)$ is a subtle matter, its delicacy being embedded in the  structure of real numbers. We may recall delicacies like: no real number nearest to zero, no smallest neighbourhood of the point $0$. Now observe that $\{(1/n, 1) :\, n \geq 2\}$ is a collection of open intervals that contains $(0, 1)$. Even $ \{(1/n, 1) : n \geq 200 \}$ sums up to $(0, 1)$, showing the superfluous role of $ \{(1/n, 1) : 2 < n < 200 \}.$ Even $\{(1/n, 1) : n \geq 2000 \}$ sums to $(0, 1)$, showing that many more are superfluous. Indeed no particular element in the covering here is indispensable. And most ridiculous, it might appear, is our inability here to specify a part (or full) of the covering without including a lot many superfluous elements. The important thing to be observed here is that in the collection $\{(1/n, 1) : n \geq 2 \}$, by discarding off the superfluous ones, we are not able to manage to cover $(0, 1)$ by a finite number of them.


 What characterises the wildness of $(0, 1)$ [not shared by $[0, 1]$] is the possibility of covering it by open intervals, in which the removal of possible superfluous ones does not reduce the covering to a finite number.

The absence of this wildness (namely the possibility of covering by open sets of the set in which the removal of possible superfluous ones, does not reduce it to finiteness) is called compactness.

 To sum up:

• A set would be called compact, if each and every coveringof the set by open sets is essentially finite in the sense that  except a finite number of the covering members all others are seen  to be superfluous.
• A set would fail to be compact, if the set provides for thepossibility of a covering by open sets, which by the removal of  superfluous members cannot be reduced to finite.

The compactness of $[0, 1]$, a land mark result of Mathematics is known as Heine-Borel Theorem. Its rigorous proof, you must observe, requires the completeness axiom of the real line, no hole in the ice-cream cup.