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What word properly completes the phrase

the radius of convergence does not depend on the $\text{______}$ of the interval

to mean that it doesn't matter whether $(a, b)$, $[a, b)$, $(a, b]$, or $[a, b]$ is the correct answer?

  • Openness and closedness don't really seem to work because the interval doesn't have to be either (it could be half-open, or, in $\mathbb{R}^n$, include any subset of its limit points).
  • Strictness makes sense, because you can say that $2$, and not $3$, is "strictly between" $1$ and $3$. However, this only really makes sense (to me) once you know the meaning; if I saw the word strictness I wouldn't really know what it meant.
  • Boundary and $endpoints$ don't work because the boundary does matter—we care what $a$ and $b$ are, just not whether they're included in the interval.

This is for a Calculus II class, so topology, etc. are outside the scope of the curriculum.

Thoughts?

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  • $\begingroup$ @BrianFitzpatrick I don't think that works because it seems to imply that we don't care what $a$ and $b$ are. $\endgroup$ – wchargin May 13 '14 at 0:35
  • $\begingroup$ Maybe "half-open/closedness" then? $\endgroup$ – Brian Fitzpatrick May 13 '14 at 0:37
  • $\begingroup$ "Half-open/closedness" is close but a bit too specific to $\mathbb{R}^1$. This is for a Calculus II class, so "topology" wouldn't be understandable. $\endgroup$ – wchargin May 13 '14 at 0:40
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    $\begingroup$ Yes but your context is the interval of convergence of a power series. There's no reason to consider anything other than $\Bbb R^1$. $\endgroup$ – Brian Fitzpatrick May 13 '14 at 0:47
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    $\begingroup$ I might use the word 'type', or I would just rewrite the sentence (the radius of convergence does not depend on whether the interval includes the endpoints or not). $\endgroup$ – Alex Zorn May 13 '14 at 0:53
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I would use:

the radius of convergence does not depend on the nature of the interval, i.e.,whether it is closed, open, or neither.

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"The radius of convergence does not depend on whether the interval is open, closed, or neither."

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