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All neighbourhoods of 1 in R contain elements that are not in (0,1]. However, in [-1,1] this is not the case as there are no elements of [-1,1] that are not in a neighbourhood of 1.
At the other end of the interval (0,1], the end is open. Every point near 0 has a neighbourhood completely contained in (0,1].
So every point of (0,1] has a neighbourhood entirely within (0,1] except 1, for which see above.