# Open Subset of a Topological Space

Let be a topological space, then a member of is said to be an open set in . Thus, in a topological space , the members of are said to be open subsets of . Since and full space are always the member of , so and are always open sets in .

On the other hand we can define as let is the topological space, then the subset of is said to be an open set of (or in ), if .

**Example:** If with topology , then and are the possible open sets of .

On the other hand if , then is not an open set of . It is clear from this illustration that the open subsets of a space depend upon the topology defined on .

**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 is an open subspace of a topological space , then each open subset of is also open in .

• Every subset of a topological space is open if and only if its each singleton subset is open.