There are actually more than a definition of the open set in topology. They are:
A set containing every interior point.
A set containing a point along the region such that you can form the open ball.
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Yes.
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No, the tournament committee set the pins. In the US Open, the
USGA set the pins, in The Open the R&A set the pins.
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f(x) = x^{2} is a continuous function on the set R of real
numbers, and (-1, 1) is an open set in R, but f(-1, 1) = [0, 1),
and [0, 1) is not an open set in R. So, f is not an open function
on R.