Boundary of Set A={a,b} in X={a,b,c} within the Topology {X, Ø, {a}, {a,b}}

In this exercise, we need to determine the boundary (\(\partial A\)) of the set \(A = \{a, b\}\) in \(X = \{a, b, c\}\) with the topology \(\{X, \{a\}, \{a, b\}, \emptyset\}\).

In this topology, the open sets are \(\{X, \{a\}, \{a, b\}, \emptyset\}\).

The closed sets are the complements of these open sets with respect to \(X = \{a, b, c\}\), specifically \(\{X, \{b,c\}, \{c\}, \emptyset\}\).

The boundary of the set \(A\) is the difference between its closure and its interior.

$$ \partial A = \text{Cl}(A) - \text{Int}(A) $$

The closure of \(A\) is the smallest closed set containing \(A = \{a, b\}\).

The smallest closed set containing \(\{a, b\}\) is \(X\), or \(\{a, b, c\}\).

$$ \text{Cl}(A) = \{a, b, c\} $$

The interior of \(A\) is the union of all open sets contained within \(A\).

In this topology, the open sets contained in \(\{a, b\}\) are \(\{a\}\) and \(\{a, b\}\).

$$ \text{Int}(A) = \{a\} \cup \{a, b\} $$

$$ \text{Int}(A) = \{a, b\} $$

Now, we can find the difference between the closure and the interior of \(A\):

$$ \partial A = \text{Cl}(A) - \text{Int}(A) $$

$$ \partial A = \{a, b, c\} - \{a, b\} $$

$$ \partial A = \{c\} $$

Therefore, the boundary of \(A = \{a, b\}\) in \(X = \{a, b, c\}\) with the topology \(\{X, \{a\}, \{a, b\}, \emptyset\}\) is \( \partial A = \{c\} \).

Alternative Method

The boundary of a set $A$ is found by intersecting the closure of $A$ with the closure of its complement $X - A$

\[ \partial A = \text{Cl}(A) \cap \text{Cl}(X - A) \]

We already know that the closure of $A$ is

$$ \text{Cl}(A) = \{a, b, c\} $$

The complement of $A = \{a, b\}$ is the set difference $X - A$

$$ X - A = \{a, b, c\} - \{a, b\} $$

$$ X - A = \{c\} $$

The closure of $X - A = \{c\}$ is the smallest closed set containing $ \{c\} $

In this topology, the closed sets are \(\{X, \{b, c\}, \{c\}, \emptyset\}\), so the smallest closed set containing $ \{c\} $ is simply $ \{c\} $

$$ \text{Cl}(X - A) = \{c\} $$

Now we can determine the boundary by intersecting the two closures:

$$ \partial A = \text{Cl}(A) \cap \text{Cl}(X - A) $$

$$ \partial A = \{a, b, c\} \cap \{c\} $$

$$ \partial A = \{c\} $$

Therefore, the boundary of $A$ is $ \partial A = \{c\} $

And so forth.