Is The Set {cos1,cos2,cos3,cos4,...} under usual topology closed??? whatbiis its closure????

To determine if the set {cos1, cos2, cos3, cos4, ...} is closed under the usual topology, we need to check if it contains all its limit points. The limit points of a set are the points to which every convergent sequence in the set converges.

In this case, we have a set of cosine values, {cos1, cos2, cos3, cos4, ...}. By observing this set, we know that each term in the sequence is bounded between -1 and 1, as the range of the cosine function is [-1, 1]. Additionally, we can see that there are infinitely many distinct values in the set.

Since the set contains infinitely many distinct values, we can construct a sequence that converges to any point in the range [-1, 1]. For example, if we want to find a sequence that converges to 0, we can consider the sequence {cos(1), cos(2π), cos(3π), ...}, where each term is chosen to be an integral multiple of π. This sequence converges to 0, as cosine is periodic and has a value of 1 at multiples of 2π. Similarly, we can construct sequences that converge to any other point in the range [-1, 1].

Since the set contains all its limit points, it is closed under the usual topology.

The closure of a set is defined as the set itself along with all its limit points. In this case, the closure of the set {cos1, cos2, cos3, cos4, ...} includes all the elements in the set as well as all the limit points. As we saw earlier, the limit points of the set are all the values in the range [-1, 1]. Therefore, the closure of the set is the set itself along with all values in the range [-1, 1], i.e., the closure is [-1, 1].