X is usually denoted (xα), where α is an index variable ranging over A. Every sequence is a net, taking A to be the directed set of natural numbers with the usual ordering. Normal A space is normal if any two disjoint closed sets have disjoint neighbourhoods. Every normal space admits a partition of unity. Normal Hausdorff A normal Hausdorff space (or T4 space) is a normal T1 space. (A normal space is Hausdorff if and only if it is T1, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff. Nowhere dense A nowhere dense set is a set whose closure has empty interior. 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". P Paracompact A space is paracompact if every open cover has a locally finite open refinement. Paracompact implies metacompact. Paracompact Hausdorff spaces are normal. Partition of unity A partition of unity of a space X is a set of continuous functions from X to [0, 1] such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1. Path A path in a space X is a continuous map f from the closed unit interval [0, 1] into X. The point f(0) is the initial point of f; the