GRE Mathematics by O. Agrawal & T. Elsner & J. Robertson & J. W. Samples & J. T. Wilson

Author:O. Agrawal & T. Elsner & J. Robertson & J. W. Samples & J. T. Wilson
Language: eng
Format: epub
ISBN: 9780738667157
Publisher: Research & Education Association
Published: 2012-09-05T00:00:00+00:00

The above properties of the neighborhood system may be used to define a topology on X.

Convergent Sequence

A sequence {a1, a2, ...} of points in a topological space X converges to a point a ∈ X if, for each open set U containing a, there exists a positive integer n0 ∈ N such that

n > n0 ⇒ an ∈ G.

Point a is called the limit of the sequence (an), denoted by

an → a or lim an = a.

Let {Tω| ω ∈ Ω } be a family of topologies on a non-empty set X. For any T1 and T2 ∈ {Tω} , if T1, ⊂ T2, we say that T1, is coarser than T2 and T2 is finer than T1, Two topologies are not comparable if neither is finer than the other.

Subspace, Relative Topology

Let A be a non-empty subset of a topological space (X, T). The class TA of all intersections of A with T-open subsets of X is a topology on A.

TA is called the relative topology on A or the relativization of T to A. The topological space (A, TA) is called a subspace of (X, T). It is easy to show that TA is actually a topology on A.

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