Connectedness: An introduction

One-piecedness and many-piecedness.

Look at the globe: Australia, Africa, North America are parts apart. So was earlier Pakistan with its East and West parts. West Indies \ comprises of so many pieces (islands) apart. On the real line, a subset $(-\infty, 5) \cup (6,\infty)$ looks like comprising of two continents.

Two continents type figure.

Likewise, the set $[3,7) \cup (17, 21) \cup \{22\} \cup \{23\}$ is like four islands.

Four islands type figure.

In contrast the sets $[1, 10]$, $(-3, 100)$, $(-\infty, \infty)$ are single pieces like one laddoo or like one chocolate or like me, like you. You may better check your single piecedness! Check also the single piecedness of your chair, your table, your glass and the plates. How do we characterise the many-piecedness in contrast to single-piecedness?

Most tempting characterisation suggested is: a model is non-single piece (many-pieced) if it can be as the union of two disjoint sets. This, however makes even $[0,4] = [0,2] \cup (2,4]$ non-single. The two sets $[0, 2]$ and $(2, 4]$ are disjoint, but even an ant would find easy to step from one part to the other. The two though disjoint don't make two islands. Why? The set $(2, 4]$ though disjoint from $[0,2]$ has a friendly point $2$ in the other piece, binding the two pieces together. Indeed, every neighbourhood of $2$ has portions of $(2,4]$. And this leads us to modify the earlier suggested characterisation of many-piecedness. We must ensure that in the two disjoint pieces comprising the model, friendly points of either of them must not be in the other. Then only, there would be at least two islands, at least two-pieces in the model. The presence of two or more disjoint pieces of this kind (separated) in a model is termed the disease of `disconnectedness' for the model.


An island is an entire world in itself. It is sovereign for all its ways and traditions, no compulsion to be consistent with traditions elsewhere and yet no conflict. As such, if a model has three pieces, a mapping on the model with values (Red), $17$ (Blue), $13000$ (Black) on the three pieces respectively is continuous.

An island is a Universe of its own; and so it is a neighbourhood of each of its points. And so each island or piece is on open set in the model. Also each island is seen to be `closed', because no outside member can be friendly to the island. This apparently funny property of being `closed' as well as `open' for a part in a model is a symptom of the disease, the disease of `disconnectedness'. More you find the symptom, the disease is greater. Suppose in $X = \{a_1, a_2, a_3,\ \ldots,\, a_6\}$, the open sets are $\phi$, $\{a_1\}$, $\{a_2\}$, $\ldots$, $\{a_6\}$, and their unions. That is to say each $a_1$, $a_2$, ..., $a_6$ is as self-centred as I (as $a_1$) was in the article ``The concept of nearness''. Then $X$ with this topology is badly disconnected. $X$ is indeed as many pieces as the number of points in it. More the selfishness in members, greater the chance of disconnectedness (or disintegration of the society). In the example in the article ``The concept of nearness'', in spite of the self-centredness of $a_1$, $X$ is not disconnected, because $a_2$, $a_3$, ..., $a_6$ are all friendly points of $\{a_1\}$, the children's love, wife's loyalty saves the family and counteracts the selfishness of $a_1$. Don't you see it is topology that determines the disease of `disconnectedness'? No gap axiom guarantees, the connectedness of $\mathbb R$.