# 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.