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1, 2.

Freddy is drawing eyeballs.

2½, 3.

One iris got bigger.

3½, 4.

The other got bigger.

5, 6.

Aura passes from and to irises.

[NUMBERS REDACTED]

Never sleep again.

What is Freddy singing about?

Subtle Hint:

Start with dots.

Moderate Hint:

The numbers are redacted because currently, numbers end by 6.

Decisive Hint:

The "aura" may seem like equipotentials between heterogeneously charged conductors: equipotentials

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4
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He is singing about

The separation axioms.

1, 2. Freddy is drawing eyeballs.

In $T_2$ spaces, or Hausdorff spaces, every two distinct points can be seperated by open sets. It is commonly depicted by the following diagram: Hausdorff space

2½, 3. One iris got bigger.

In $T_3$ spaces, or regular spaces, every disjoint closed set and point can be seperated by open sets. It is commonly depicted by the following diagram:regular space

3½, 4. The other got bigger.

In $T_4$ spaces, or normal spaces, every two disjoint closed sets can be seperated by open sets. It is commonly depicted by the following diagram:normal space

5, 6. Aura passes from and to irises.

In $T_6$ spaces, or perfectly normal spaces, for every two disjoint closed sets, there exists a continuous function whose return value is 0 on and only on one closed set and 1 on and only on the other closed set. (Hence the decisive hint)

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  • $\begingroup$ Great! This probably should be the accepted answer (regarding the existence of $T_{2\frac12}$ and $T_{3\frac12}$). $\endgroup$ – trolley813 Jan 6 at 22:08
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    $\begingroup$ @trolley813 Well, it is the OP self-answering - I'd very much hope this was correct! :) $\endgroup$ – Stiv Jan 6 at 22:25
  • $\begingroup$ For those of us too slow to make the connection ourselves, can you add an explanation of how Freddy Krueger fits in? $\endgroup$ – tmpearce Jan 9 at 16:34
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Maybe he is singing about

Cassini ovals? (or some similar curves)

Reasoning

The equation of the oval is $(x^2+y^2)^2-2a^2(x^2-y^2)+a^4=b^4$. When the $c = \dfrac b a$ ratio increases, the oval initially looks like "eyeballs" (2 separate loops, indeed starting with 2 dots when $c=0$), with increasing "irides", then (when $c=1$) the halves are fusing with each other (it will be Bernoulli lemniscate - an 8-shaped curve), allowing the "aura" to pass between them, and next (when $c>1$) turning into single peanut-shaped (later, when $c>\sqrt2$, an oval-shaped) curve.

However

this answer bears absolutely no connection with A Nightmare on Elm Street franchise, and it probably has to have one (since Freddy Krueger is the protagonist name, and Never Sleep Again is a documentary film that chronicles the entire franchise, and the number of installments in the main series is 6 at the moment (currently numbers end by 6!)).

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  • $\begingroup$ Ha, nice and creative guess, but not what I intended. I intended some other branch of math. You should consider why there are numbers like 2 1/2. $\endgroup$ – Dannyu NDos Dec 30 '19 at 7:12
  • $\begingroup$ Hint: I didn't intend much connection to the franchise. $\endgroup$ – Dannyu NDos Dec 30 '19 at 7:13
  • $\begingroup$ As your answer was the only one, I gave you the bounty. $\endgroup$ – Dannyu NDos Jan 5 at 20:17
  • $\begingroup$ @DannyuNDos Thanks! It's a pity that nobody gave a better answer. $\endgroup$ – trolley813 Jan 5 at 21:36

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