# 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.