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In this post on Reddit, a user proposes an alternate definition of compactness, as an "induction principle":

A topological space $X$ is compact iff given a statement $P$ whose truth or falsity is defined on open subsets of $X$, we have $P(X)$ as soon as

  1. For all $x \in X$ there exists open $U \ni x$ such that $P(U)$.

  2. $P(U_1), \dotsc, P(U_n) \implies P(U_1 \cup \dotsb \cup U_n)$.

To me this seems like such a drastically better definition of compactness: more intuitive, more mathematical, more to the point. The statement in terms of open covers is useful in proofs, but it should be secondary to this.

Any thoughts/opinions? Why don't we teach compactness this way, except that that's the way it's always been done?

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    $\begingroup$ "Any thoughts/opinions?" question should be avoided here, but the second part is alright. Additionally, "pedagogically terrible" is a bit of a stretch. I think a better title is "Teaching an alternate definition for a compact topological space via the induction principal?" $\endgroup$ – Chris C Mar 24 '15 at 21:13
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    $\begingroup$ Guys, I don't think Eric is saying to never introduce the standard definition. Just use this definition first, and develop the standard one as equivalent. $\endgroup$ – Steven Gubkin Mar 24 '15 at 22:25
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    $\begingroup$ Also, as a technical nitpick: first-order logic has a compactness theorem while second-order logic doesn't. In order for this to be used properly, one has to exercise care on the properties P for which the statement is being used. I think the posting gives a good first attempt and may lead to some good intuition, but without the appropriate technical restrictions it may also lead many astray. In particular, is X a set of members that don't contain themselves? Gerhard "Ill-Foundedness: Don't Get Me Started" Paseman, 2015.03.24 $\endgroup$ – Gerhard Paseman Mar 24 '15 at 23:27
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    $\begingroup$ @GerhardPaseman Perhaps your thoughts could be organized into a good answer for the question. $\endgroup$ – Chris C Mar 25 '15 at 1:18
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    $\begingroup$ When first proposed in the early 1900's, the Heine-Borel theorem was thought of as "continuous induction" for an interval of real numbers. Also see: math.stackexchange.com/a/4424/442 $\endgroup$ – Gerald Edgar Mar 25 '15 at 12:32

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