A small tale with a topological soul, with the aim of providing a very high level intuition for the notion of density and dense set in topology.
Smally Open was a cheeky youngster of the Open family who lived in a large meadow. It was a very nice and green meadow, well cared for and abundant in clovers. Smally Open liked it much because he found it magic. Oh, I wish you were there as well, my smally readers: Smally Open climbed the highest trees, went as far as he could from his home, went down below the blades of grass, but there was no way: anyway he was, Smally Open would always see his verdant meadow, as if it moved with him and changed its look. Anyway he looked, Smaly Open would always see a large green meadow.
You will wonder, my dears, if Smally Open did not get bored to death in always seeing the very same meadow anywhere he went… well, I hope you do not believe that it was just all alike! The meadow looked all the same if you looked from afar and without attention, but sometimes there was a butterfly here, some other time there was an amazingly beautiful red leaf there: at any rate, there was always some new thing to fascinate him.
Anyway, when Smally Open did get bored, there always was a nice way he liked to use to spend his time: go and bother Mr. Dense-D. He was pretty a weird and extraordinary sir, in fact he had a lot of heads (how many, you ask? Try to count the stars in a cloudless night: Mr Dense-D had at least three-times-twice-one-thousand-times-that-number-of-heads!) You should not however think of D-Dense as a green weird martian with several scary heads, going away pecking at children’s ears, no! D-Dense was a very distinguished man, very kind and happy.
Indeed, D-Dense was the happiness of the meadow in which he lived together with Smally Open, because he was its clovers. But not a few crumpled and frail clovers, oh well stop with these prejudices towards Mr. D-Dense! No, those were beautiful clovers, big and comfortable, and above all there was one in any part of the meadow you would look for. When Smally Open was tired, he would look around and immediately find one of the heads of Mr. D-Dense, upon which he could rest under the sunlight.
Apertolo, paying, would often jump from one blade of grass to another. When he would lose his balance and fall in the void he was always lucky that Mr. D-Dense lived in his same meadow, because were it not for his many heads that would catch him, falling on the ground he would at least get a big bruise! But Smally Open sometimes take advantage of Mr. D-Dense’s kindness: he would jump on his heads from tree tops on purpose, he would eat on the clovers, crumbling on D-Dense’s heads, or he would ruffle all his hair intentionally. At any rate, he would really bother him! Mr. D-Dense was a patient man, but one day he was fed up and said: “Enough, I am old and tired: I need a quiet holiday. I am going to leave.” And so he started lifting himself.
First, Smally Open fell on the ground. Then he heard a great rumble, as if the whole land was shaking, and saw all the clovers moving and lifting, with mountains of soil up in the air, until Mr. D-Dense headed with all his clover-heads towards the horizon.
Days went by since Mr. D-Dense had left, and Smally Open did not know what to do. The meadow was not as nice as before: now small holes could be seen everywhere, left by Mr. D-Dense’s heads, and, from any perspective he looked the meadow at, it now looked sad and battered, as if something was missing and it was not the magic meadow he so much liked anymore. Moreover, when Smally Open was tired, he could but sit on the ground, in the shade: he missed so much the comfortable and cozy clovers couch. To have fun again, he tried to jump from one blade of grass to another, but this time, when he fell, Mr. D-Dense was not there to grab him and he got himself a big bruise. Smally Open was not able to build anything nice and funny as when Mr. D-Dense was his playmate.
By now, Smally Open spent several hours on the top of the highest tree, with the hope of seeing Mr. D-Dense coming back. Just when he was about to lose all hopes, there he caught a glimpse of a shadow, far far away. It just looked the shadow of someone with lots of heads… and who could it have been if not Mr. D-Dense? Smally Open got down the tree and run towards him. Indeed, it was Mr. D-Dense! Smally Open went closer to one of his heads, patted a delicate knock-knock, and said: “I am sorry Mr. D-Dense for jumping on your heads and for eating on your heads and for ruffling your hair, I will not do it again. But will you come back? Mr. D-Dense took Smally Open on one of his clover heads and told him: “Of course I will come back. Come, I have got to tell you about the friends I met again in this holiday of mine: the waves of the sea, the dunes of the desert, the stars of the sky…”