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Topology

  1. If \(X\) is a topological space with topology \(\mathscr T\), we say that a subset \(U\) of \(X\) is an open set of \(X\) if \(U\) belongs to the collection \(\mathscr T\). Using this terminology, one can say that a topological space is a set \(X\) together with a collection of subsets of \(X\), called open sets, such that \(\varnothing\) and \(X\) are both open, and such that arbitrary unions and finite intersections of open sets are open.

  2. If \(X\) is a set, a basis for a topology on \(X\) is a collection \(\mathscr B\) of subsets of \(X\) (called basis elements) such that

  1. For each \(x \in X\), there is at least one basis element \(B\) containing \(x\).
  2. If \(x\) belongs to the intersection of two basis elements \(B_1\) and \(B_2\), then there is a basis element \(B_3\) containing \(x\) such that \(B_3 ⊂ B_1 ∩ B_2\).
    If basis \(\mathscr B\) satisfies these two conditions, then we define the topology \(\mathscr T\) generated by \(\mathscr B\) as follows: A subset \(U\) of \(X\) is said to be an open subset in \(X\) (that is, to be an element of \(\mathscr T\)) if for each \(x \in U\), there is a basis element \(B \in \mathscr B\) such that \(x \in B\) and \(B \subset U\). Note that each basis element is itself an element of \(\mathscr T\).
  1. If \(X\) is any set, the collection of all subsets of \(X\) is a topology on \(X\); it is called the discrete topology. The collection consisting of \(X\) and \(\varnothing\) only is also a topology on \(X\); we shall call it the indiscrete topology, or the trivial topology.

BIBLIOGRAPHY

1. Munkres JR. Topology. Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9.; 2000.