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 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 in X. The subset \left\{ 2 \right\} is also neither open nor closed in X.

In general, 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 clopen set. Since \phi and X are open as well as closed, so there are clopen sets. Since each subset of a discrete topological space is open as well as closed so 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.

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