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.