Cardinal Function - Cardinal Functions in Topology

Cardinal Functions in Topology

Cardinal functions are widely used in topology as a tool for describing various topological properties. Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology", prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the definitions given below, e.g. by adding "" to the right-hand side of the definitions, etc.)

• Perhaps the simplest cardinal invariants of a topological space X are its cardinality and the cardinality of its topology, denoted respectively by |X | and o(X).
• The weight w(X ) of a topological space X is the cardinality of the smallest base for X. When w(X ) = the space X is said to be second countable.
  • The -weight of a space X is the cardinality of the smallest -base for X.
• The character of a topological space X at a point x is the cardinality of the smallest local base for x. The character of space X is When the space X is said to be first countable.
• The density d(X ) of a space X is the cardinality of the smallest dense subset of X. When the space X is said to be separable.
• The Lindelöf number L(X ) of a space X is the smallest infinite cardinality such that every open cover has a subcover of cardinality no more than L(X ). When the space X is said to be a Lindelöf space.
• The cellularity of a space X is is a family of mutually disjoint non-empty open subsets of .
  • The Hereditary cellularity (sometimes spread) is the least upper bound of cellularities of its subsets: or with the subspace topology is discrete .
• The tightness t(x, X) of a topological space X at a point is the smallest cardinal number such that, whenever for some subset Y of X, there exists a subset Z of Y, with |Z | ≤, such that . Symbolically, The tightness of a space X is . When t(X) = the space X is said to be countably generated or countably tight.
  • The augumented tightness of a space X, is the smallest regular cardinal such that for any, there is a subset Z of Y with cardinality less than, such that .