# 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 members of $$\tau $$, so $$\phi $$ and $$X$$ are always open sets in $$X$$.

On the other hand we can define it as let $$\left( {X,\tau } \right)$$ be 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 number 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.

arofenitra

February 21@ 4:13 amHow to prove that “Every subset of a topological space is open if and only if its each singleton subset is open”?