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poutsos.A
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Given that f is a function from R(=real Nos) to R continuous on R AND ,A any subset of R,IS THE closure of f(A) ,a closed set??
poutsos.A said:Yes, you right thank you. But if we define a set to be closed if its complement is open,
how then we prove its closure to be a closed set??
A closed set is a set of points in a topological space that includes all of its limit points, meaning that every sequence of points in the set that converges also converges to a point in the set.
The closure of a set is defined as the smallest closed set that contains all points in the original set. It is the union of the original set and all of its limit points.
The closure of a set contains all of its limit points, meaning that the set includes all points that a sequence can approach as it converges. However, not all limit points are necessarily in the closure of the set.
A set is closed if it contains all of its limit points. One way to determine this is by checking if the set is equal to its closure. If the closure of the set is equal to the set itself, then the set is closed.
In general, a set cannot be both open and closed. However, in some cases, a set can be both open and closed if it is the empty set or the entire space. In other words, a set is either open or closed, but not both.