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The title says all..

There are properties such that if property is true at a point, then it is true around a small neighborhood as well. I could have sworn that there is a word for it, but I can't remember it for the life of me.

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    "Open"? (That's certainly the right word to describe the set of points with the property.) Or perhaps "semidecidable"? (See What concept does an open set axiomatise?.) – Calum Gilhooley Sep 16 '21 at 14:54
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    I've never heard of this kind of this being in reference to a property itself, but always in reference to the set of points that have the given property, in which case you are just referring to an open set. I have heard of a similar thing about being able to determine the property only given a small neighborhood of information, in which case we refer to it as a "local" property, e.g. differentiability is local, but I've never heard "open" used to describe that the set on which the property is true is open. – nullUser Sep 16 '21 at 15:13
  • Interior point? –  Sep 16 '21 at 16:31
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    The term "locally" is often used when a property is assumed to exist at all points in a neighborhood of each point -- locally continuous, locally constant, etc. Although this doesn't answer the question you actually asked, it might answer the question you intended to ask. – Dave L. Renfro Sep 16 '21 at 17:25
  • Continuity of the property? – herb steinberg Sep 16 '21 at 17:26
  • What kind of property you have in mind? A point is such a simple space that it satisfies most interesting properties (compact, connected, path connected, simply connected, contractible, etc) regardless of its neighbourhood. Such definition doesn't seem to be useful in general. The only interesting example I can think of is that if $f:X\to\mathbb{R}$ is continuous and nonzero at some point then it is nonzero at some neighbourhood of that point. This doesn't have a name though and is a rather straightforward consequence of continuity. – freakish Sep 16 '21 at 18:18
  • @freakish: The only interesting example I can think of is --- Another example is being analytic at a point, although in this case the result is almost immediate from the definition, unlike the case for the example you gave. – Dave L. Renfro Sep 17 '21 at 06:42

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