Real induction is a useful proof technique which can be thought as a version of "continuous" induction. I will include here version from Pete L. Clark's text mentioned in this answer,1 where it is formulated like this:2

Let $a < b$ be real numbers. Let a subset $S \subseteq [a,b]$ be inductive, i.e.:
(RI1) $a\in S$.
(RI2) If $a\le x<b$, then $x\in S$ $\implies$ $[x,y]\subseteq S$ for some $y > x$.
(RI3) If $a < x \le b$ and $[a,x)\subset S$, then $x \in S$.
Then $S=[a,b]$.

Stated informally (and a bit vaguely and imprecise), if I want to prove that some property holds for all points of the interval $[a,b]$, I can prove this "from left to right" by checking that:

  • The leftmost point $a$ has the required property.
  • If $x$ has the given property, then I can show that at least some points near $x$ to the right have this property.
  • If I can find points with this property arbitrarily close on left from $x$, then also $x$ has this property.

(This does not correspond exactly to the three conditions above, but if you consider that the property we are trying to prove is "$[a,x]\subseteq S$", then the correspondence between these three bullet points and the conditions (RI1), (RI2), (RI3) seems to be more natural.)

The linked text also discusses some generalization of this principle to linearly ordered sets and ordered fields. There is also this question on MathOverflow which asks about generalization to partial orders: A principle of mathematical induction for partially ordered sets with infima?

I wonder whether we can generalize this to some context where we do not have ordering.

Question. Are there some natural generalizations of the above method to topological spaces or metric spaces? Are there some references where they are studied? Are there some interesting applications?

When I tried to mimic the three above conditions, I was only able to come up with this.

If $X$ is connected then we get $S=X$ for any set which fulfills these conditions:

  1. $S\ne\emptyset$.
  2. If $x\in S$ then there exists an open set $U$ such that $x\in U\subseteq S$.
  3. If $x\in\overline S$, then $x\in S$.

The second condition only says that $S$ is open. The third condition basically says that $\overline S=S$, i.e., that $S$ is closed. But I wanted to formulate it in such way that it resembles the formulation of real induction given above.

This is a very naive version, since it only says that the only non-empty clopen subset of a connected space $X$ is the whole space.

This version seems not very satisfactory since when we view the real induction informally (as described above) as a process in which the set $S$ is "growing" until it is the whole $x$, this is entirely lost in the above formulation. Another reason is that real induction feels a bit like the correct generalization should also be related to compactness.

I could try also to define some sets by transfinite induction:

  • $S_0$ is a singleton.
  • $S_{\alpha+1}$ is an open neighborhood os $\overline{S_\alpha}$.
  • If $\alpha$ is a limit ordinal, then $S_\alpha=\bigcup\limits_{\beta<\alpha} S_\beta$.

The claim would than be that there is an $\alpha$ such that $S_\alpha=X$. (For example, if $X$ is connected.)

This seems a bit closer to my informal description above in the sense, that we have some set which is "growing". But it seems to be unnecessarily complicated way to write more-or-less the same thing without actually gaining anything.

So I still wonder whether there is some reasonable generalization, which would resemble real induction and be actually useful. (Although I fear without some kind of partial order or at least pre-order we have changed too much to expect something nice and useful.)

1 Pete L. Clark: The Instructor's Guide to Real Induction, https://arxiv.org/abs/1208.0973, http://math.uga.edu/~pete/realinduction.pdf
2 In fact, in this text it is shown that $S$ is inductive if and only if $S=[a,b]$. However, the other implication is obvious. I have only included the direction which is relevant for correctness of this proof technique.

  • $\begingroup$ I think that your connectedness formulation is actually very natural, and it certainly is extremely useful (it is basically what you do every time you use connectedness of a space). Also, a related question: math.stackexchange.com/questions/1357939/… $\endgroup$ – Eric Wofsey Apr 21 '17 at 17:25

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