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Is the image of an open set under a continuous mapping need not be open?
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.
In calculus do open sets have accumulation points?
Yes, every point in an open set is an accumulation point.
What does it means for a set of numbers to be open?
It means that the boundaries of the set are not included in the set. For example, consider the set of numbers that are bigger than 1 and smaller than 2. The set is bounded by 1 and 2 but neither of these belong to the set.
Why empty set is open on the basis of def?
The empty set is open because the statement: "if x in A, some neighborhood of x is a subset of A" is true! If A is empty, the hypothesis: "if x in A" is false and so the statement is vacuously true.
What is an open circle in math?
It is the set of all point INSIDE the circle but not points on the circumference.
