# Closed Subset of a Topological Space

Let $\left( {X,\tau } \right)$ be a topological space, then a subset of X whose complement is a member of $\tau$ is said to be a closed set in X. Thus in a topological space $\left( {X,\tau } \right)$, the complements of the members of $\tau$ are said to be closed subsets of X since $\phi$ and the full space X are always closed sets of X.

On the other hand we can define this as let $\left( {X,\tau } \right)$ be a topological space, then the subset $A$ of X is said to be closed in X if ${A^c} \in \tau$ (${A^c}$ is open in X.

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 closed subsets of X.

Remark:

If $X = \left\{ {1,2,3,4} \right\}$ with topology $\tau = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 4 \right\},\left\{ {1,4} \right\},X} \right\}$, the subset $\left\{ 3 \right\}$ of X is such that neither closed nor open set is in X. The subset $\left\{ 2 \right\}$ is also neither open nor closed in X.

In general, in the discrete topological space $\left( {X,\tau } \right)$ there does not exist any subset of X which is neither open nor closed in X, i.e. all the subsets in discrete topological space are open as well as closed.

Clopen Set

Let $\left( {X,\tau } \right)$ be a topological space. A subset of X which is open as well as closed is said to be a clopen set. Since $\phi$ and X are open as well as closed, there are clopen sets. Since each subset of a discrete topological space is open as well as closed, each subset of a discrete topological space is a clopen set.

Theorems

• Every subset of a discrete topological space is closed.
• The intersection of any number of closed subsets of a topological space is closed.
• The union of any finite number of closed subsets of a topological space is closed.
• Every subset of a discrete topological space is clopen.