In option D), you can take a single-element set for the set S, for example, say S = {3}. Then the set as described in option D) corresponds to the set R\{3}. Since {3} is a closed set, its complement is open and this complement is exactly R\{3}.
There are a few ways to define closure that are equivalent, and this isn’t usually the first one I think of or the one that’s most natural to me. I think that’s why l took a bit of time to recognize that option D had something a bit “off.”
In option D), you can take a single-element set for the set S, for example, say S = {3}. Then the set as described in option D) corresponds to the set R\{3}. Since {3} is a closed set, its complement is open and this complement is exactly R\{3}.
the option D is really close to the definition of the closure, the only difference being the W intersect S = 0 instead of W intersect S != 0
There are a few ways to define closure that are equivalent, and this isn’t usually the first one I think of or the one that’s most natural to me. I think that’s why l took a bit of time to recognize that option D had something a bit “off.”