How do you show a space is closed?

How do you show a space is closed?

To prove that A is closed, it suffices to prove that if (xn) is a sequence of points of A which converges to x∈S, then x∈A. So let xn be such a sequence. Since xn converges in S, it is a Cauchy sequence in S. Therefore (xn) is also a Cauchy sequence in A, so by completeness of A, there exists some y∈A such that xn→y.

How do you prove a subset of a metric space is open?

Theorem A3 A subset U of a metric space (X, d) is open if and only if it is the union of open balls. [0, 1). However, if [0, 1) is considered to be the entire space X, then it is open by Theorem A2(a). If U is an open subset of a metric space (X, d), then its complement Uc = X – U is said to be closed.

How do you prove something is closed under multiplication?

A set is closed under addition if you can add any two numbers in the set and still have a number in the set as a result. A set is closed under (scalar) multiplication if you can multiply any two elements, and the result is still a number in the set.

Is a subset of a closed set closed?

A subset A of a topological space X is said to be closed if the set X – A is open. Theorem 1.2. Let Y be a subspace of X . Then a set A is closed in Y if and only if it equals the intersection of a closed set of X with Y .

How do you prove a subset of R 2 is closed?

In R2 a set is closed if it contains all of its limit points. So, you can prove C is closed by considering a sequence in C and show that if it converges then the limit is in C. More generally, if f:R2→R is a continuous function then the set {(x,y)∈R2∣f(x,y)=c}, for any constant c∈R, is closed.

Is the closure of any open ball in any metric space the closed ball with the same center and radius?

For any metric space (X,d), the following are equivalent: For any x∈X and radius r, the closure of the open ball of radius r around x is the closed ball of radius r.

What is a closed ball?

A closed n-ball of radius r is the set of all points of distance less than or equal to r away from x. In Euclidean n-space, every ball is bounded by a hypersphere. The ball is a bounded interval when n = 1, is a disk bounded by a circle when n = 2, and is bounded by a sphere when n = 3.