Glossary of Topology - O

O

Open cover

An open cover is a cover consisting of open sets.

Open ball

If (M, d) is a metric space, an open ball is a set of the form B(x; r) := {y in M : d(x, y) < r}, where x is in M and r is a positive real number, the radius of the ball. An open ball of radius r is an open r-ball. Every open ball is an open set in the topology on M induced by d.

Open condition

See open property.

Open set

An open set is a member of the topology.

Open function

A function from one space to another is open if the image of every open set is open.

Open property

A property of points in a topological space is said to be "open" if those points which possess it form an open set. Such conditions often take a common form, and that form can be said to be an open condition; for example, in metric spaces, one defines an open ball as above, and says that "strict inequality is an open condition".