Open Subset of a Topological Space

Let \left( {X,\tau } \right) be a topological space, then a member of \tau is said to be an open set in X. Thus, in a topological space \left( {X,\tau } \right), the members of \tau are said to be open subsets of X. Since \phi and full space X are always the member of \tau , so \phi and X are always open sets in X.

On the other hand we can define as let \left( {X,\tau } \right) is the topological space, then the subset A of X is said to be an open set of X (or in X), if A \in \tau .

Example: If X = \left\{ {a,b} \right\} with topology \tau  = \left\{ {\phi ,\left\{ a \right\},\left\{ b \right\},X} \right\}, then \phi ,X,\left\{ a \right\}and\left\{ b \right\} are the possible open sets of X.

On the other hand if \tau  = \left\{ {\phi ,\left\{ a \right\},X} \right\}, then \left\{ b \right\} is not an open set of X. It is clear from this illustration that the open subsets of a space X depend upon the topology defined on X.

Theorems:

• Every subset of a discrete topological space is open.
• The union of any numbers of open subsets of a topological space is open.
• The intersection of any finite number of open subsets of a topological space is open.
• If Y is an open subspace of a topological space X, then each open subset of Y is also open in X.
• Every subset of a topological space is open if and only if its each singleton subset is open.