Can a set in a metric space be open and closed?

Note that a set can be both open and closed; for example, the empty set is both open and closed in any metric space. Furthermore, it is possible for a set to be neither open nor closed; for example, in a half-open bounded interval is neither open nor closed.

What is open set in metric space?

R2 | f(x, y) < 1} with f(x, y) a continuous function, is an open set. Any metric space is an open subset of itself. The empty set is an open subset of any metric space. We will see later why this is an important fact.

What is a closed set and open set?

(Open and Closed Sets) A set is open if every point in is an interior point. A set is closed if it contains all of its boundary points.

What are the open sets of a discrete metric space?

(2) Let (X, d) be a discrete metric space. Thus singletons are open sets as {x} = B(x, ϵ) where ϵ < 1. Any subset A can be written as union of singletons. As any union of open sets is open, any subset in X is open.

Which sets are open and closed?

In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive.

How do you prove a metric space is closed?

If U is an open subset of a metric space (X, d), then its complement Uc = X – U is said to be closed. In other words, a set is closed if and only if its complement is open. For example, a moments thought should convince you that the subset of ®2 defined by {(x, y) ∞ ®2: x2 + y2 ¯ 1} is a closed set.

Is 0 an open set?

Since the point 0 cannot be an interior point of your set, the set {0} cannot be an open set.

Are singleton sets open?

Every singleton set is closed. It is enough to prove that the complement is open. Consider {x} in R. Then X∖{x}=(−∞,x)∪(x,∞) which is the union of two open sets, hence open.

Is a discrete set open or closed?

In the discrete topology any subset of S is open. In the discrete topology no subset of S other than S and ∅ are open. Note that in any topology there are at least two sets which are both open and closed, S and ∅. In the discrete topology all subsets of S are both open and closed.

Is R 2 open or closed?

This is obvious topologically (the whole space is open by definition, but it is also the complement of the (open) empty set, and so it is also closed), but there’s no need to abstract as far as topology with Rn; that every point in R2 is an interior point (has an open ball in R2) in should be obvious, so it is open.