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First and Second Countable spaces. Cover. Lindelof space. Dense set. Separable space. Hereditary property.

Def. First countable space. A topological space X is called a first countable space if it satisfies the following axiom, called the first axiom of countability.

[C1]     For each point p ε X there exists a countable class Bp of open sets containing p such that every open set G containing p also contains a member of Bp.

In other words, a topological space X is a first countable space if and only if there exists a countable local base at every point p ε X.

Example 1. Let X be a metric space. Let p ε X. The countable class of open spheres {S(p, 1), S(p, 1/2), S(p, 1/3), ..... }with center at p is a local base at p. Thus every metric space satisfies the first axiom of countability.

Example 2. Let X be any discrete space and let p ε X. A local base at point p is the singleton set {p} — which is countable. Thus every discrete space satisfies the first axiom of countability.

Theorem 1. A function defined on a first countable space X is continuous at p ε X if and only if it is sequentially continuous at p.

In other words, if a topological space X satisfies the first axiom of countability, then f : X Y is continuous at p ε X if and only if for every sequence {an} in X converging to p, the sequence {f(an)} in Y converges to f(p), i.e.

Def. Second countable space. A topological space X with topology τ is called a second countable space if it satisfies the following axiom, called the second axiom of countability.

[C2]     There exists a countable base B for the topology τ.

Example 3. The class of open intervals (a, b) with rational endpoints — i.e. a, b ε Q where Q is the set of rational numbers — is countable and is a base for the usual topology on the real line R. Thus R satisfies the second axiom of countability and is thus a second countable space.

Example 4. Consider the real line R with the discrete topology D. Now the one and only base for a discrete topology on a set X is that collection B of all the singleton sets of X. We now note that R is non-countable and thus the class of singleton sets {p} of R is non-countable. Thus the topological space R with the discrete topology D does not satisfy second axiom of countability. By the same logic, the topological space Q of rational numbers with the discrete topology D does satisfy second axiom of countability.

If B is a countable base for a space X, and if Bp consists of the members of B which contain the point p ε X, then Bp is a countable local base at p. In other words,

Theorem 2. A second countable space is also first countable.

Def. Cover. Let S be a set. Let A be a subset of S. Then a collection C of subsets of S is a cover of A if A is a subset of the union of the members of C i.e.

If each member of C is an open subset of S, then C is called an open cover of A. If C contains a countable subclass which also is a cover of A , then C is said to be reducible to a countable cover of A.

Syn. Covering

Theorem 3. Let A be any subset of a second countable space X. Then every open cover of A is reducible to a countable cover.

Theorem 4. Let X be a second countable space. Then every base B for X is reducible to a countable base for X.

Def. Lindelof space. A topological space X is called a Lindelof space if every open cover of X is reducible to a countable cover.

Thus every second countable space is a Lindelof space.

Def. Dense set. A set E in a space M is dense (or dense in M, or everywhere dense) if every point of M is a point of E or a limit point of E, or (equivalently) if the closure of E is M, or if every neighborhood in M contains a point of E. A set E is dense in itself if every point of E is an accumulation point of E; i.e. if each neighborhood of any point of E contains another point of E.

Example 5. The set of rational numbers is dense in itself and dense in the set R of all real numbers, as is also the set of irrational numbers. This is equivalent to the fact that between any two real numbers (either rational or irrational) there are both rational and irrational numbers.

James & James. Mathematics Dictionary

Def. Nondense set. A set E in a space M is nondense (or nowhere dense) relative to M if the closure of E contains no interior points, or (equivalently) if the complement of the closure of E is dense in M.

Example 6. The set S = {0, 1, 1/2, 1/3, 1/4, .....} is nowhere dense in R, since 0 is the only accumulation point of S and the closure of S contains no interior points.

Def. Separable space. A (topological) space which contains a countable (or finite) set W of points which is dense in the space; i.e. every neighborhood of any point in the space contains a point of W. A space which satisfies the second axiom of countability is separable. Such a space is sometimes said to be completely separable, perfectly separable or simply separable. Hilbert space and Euclidean space of n dimensions are separable.

James & James. Mathematics Dictionary

A topological space X is separable if and only if there exists a finite or denumerable subset A of X such that the closure of A is the entire space i.e. = X.

Example 7. The real line R with the usual topology is separable since the set Q of rational numbers is denumerable and is dense in R, i.e. = R.

Every second countable space is separable but not every separable space is second countable. For example, the real line R with the topology generated by the closed-open intervals [a, b) is a classic example of a separable space which does not satisfy the second axiom of countability.

Theorem 5. Every separable metric space is second countable.

Example 8. Let C [0, 1] denote the linear space of all continuous functions on the closed interval [0, 1] with the norm defined by

|| f || = sup{ |f(x)|: 0 x 1 }

By the Weierstrass Approximation Theorem, for any function f ε C [0, 1] and any ε > 0, there exists a polynomial p with rational coefficients such that

|| f - p || < ε

(that is:             |f(x) - p(x)| < ε          for all x ε [0,1] )

Thus the collection P of all such polynomials is dense in C [0, 1] . Now P is a countable set so C [0, 1] is separable and, by Theorem 5, second countable.

Def. Hereditary property. A property P of a topological space X is said to be hereditary if every subspace of X also possesses the property.

Theorem 6. Every subspace of a first countable space is first countable. Every subspace of a second countable space is second countable.

Thus the properties of first and second countability are hereditary properties. However, the property of separability is not hereditary.

References

1. Lipschutz. General Topology

2. James & James. Mathematics Dictionary