The closure of a subset of topological Space is the intersection of all it’s closed super sets.In others words the closure of a subset of topological Space is the smallest closed super set of a subset of topological Space.
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| What is the closure of a subset of a topological space? |
Related Questions;
In the discrete topological space (X,D), what is the closure of A where A is a subset of X?
The closure of a subset A of topological Space (X, T) is also the union of A and A^d.
Cl(A) = A ∪ A^d
Examples:
1) Let T = { Φ, X, {p, q}, {r}, {p, q, r} } be topology defined on X = { p, q, r, t} and A = {p, r} is the subset of X. Then find Cl(A).
Solution;
All closed sets are
Φ, X, {p, q}, {r}, {p, q, r}
All closed super sets of A are
X, { p, q, r}
Now Cl(A) = X ∩ {p, q, r} = {p, q, r}
2) Let T= { Φ, X, {11, 14}, {17, 20} } be topology defined on X = { 11, 14, 17, 20} and B = {11, 17, 20}. Then find Cl(B).
Solution;
All closed sets are
Φ, X, {11, 14}, {17, 20}
All closed super sets of A is
X
Now Cl(B) = X