$\underline{\mathbf{UPDATE}}$
Q = ImplicitRegion[x > 0 && x < 0, {x}]; Q = If[RegionDimension[Q] == -\[Infinity], EmptyRegion[1]] returns:
EmptyRegion[1] There is nothing wrong here:$\underline{\mathbf{UPDATE}}$
\[ScriptCapitalQ] = ImplicitRegion[x > 0 && x < 0, {x}]; Reduce[Element[{x}, \[ScriptCapitalQ]], x] This returns False as you have pointed out. This means the region is empty.
Region[\[ScriptCapitalR]] gives
whitespace (empty set)
Another example,
\[ScriptCapitalQ] = ImplicitRegion[2 > xIf[RegionDimension[\[ScriptCapitalQ]] >== 0-\[Infinity], {x}]; Reduce[Element[{x}EmptyRegion[1], \[ScriptCapitalQ]], x] returns:
0 < x < 2 and Region[\[ScriptCapitalQ]] gives
Presumably this is a segment between 0 and 2!
Also RegionQ[ \[ScriptCapitalR]] evaluates to True, so it is a region (albeit empty)!
Finally,
RegionDimension[EmptyRegion[1]] returns '-[Infinity]'
RegionQ[ \[ScriptCapitalR]]EmptyRegion[1] returns '-[Infinity]'
There is nothing wrong here:
\[ScriptCapitalQ] = ImplicitRegion[x > 0 && x < 0, {x}]; Reduce[Element[{x}, \[ScriptCapitalQ]], x] This returns False as you have pointed out. This means the region is empty.
Region[\[ScriptCapitalR]] gives
whitespace (empty set)
Another example,
\[ScriptCapitalQ] = ImplicitRegion[2 > x > 0, {x}]; Reduce[Element[{x}, \[ScriptCapitalQ]], x] returns:
0 < x < 2 and Region[\[ScriptCapitalQ]] gives
Presumably this is a segment between 0 and 2!
Also RegionQ[ \[ScriptCapitalR]] evaluates to True, so it is a region (albeit empty)!
Finally,
RegionDimension[EmptyRegion[1]] returns '-[Infinity]'
RegionQ[ \[ScriptCapitalR]] returns '-[Infinity]'
$\underline{\mathbf{UPDATE}}$
\[ScriptCapitalQ] = ImplicitRegion[x > 0 && x < 0, {x}]; \[ScriptCapitalQ] = If[RegionDimension[\[ScriptCapitalQ]] == -\[Infinity], EmptyRegion[1], \[ScriptCapitalQ]] returns:
EmptyRegion[1]