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

In this exercise, we need to determine the boundary ($\partial A$) of the set $A = \{a, c\}$ within $X = \{a, b, c\}$ using 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 the open sets: $\{X, \{b, c\}, \{c\}, \emptyset\}$.

The boundary of $A$ is the difference between the closure and the interior of the set $A$.

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

The closure of $A$ is the smallest closed set that contains $A = \{a, c\} $.

In this case, the smallest closed set containing $\{a, c\}$ is $X$, which is $\{a, b, c\}$.

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

The interior of $A$ is the union of all open sets contained within $A$.

In the given topology, the open sets are $\{X, \emptyset, \{a\}, \{a, b\}\}$.

The only open set contained within $\{a, c\}$ is $\{a\}$, so the interior of $A$ is:

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

Now, we calculate the difference between the closure and the interior of $A$:

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

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

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

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

Alternative Solution

The boundary of $A = \{a,c\}$ can also be found by taking the intersection of the closure of $A$ and the closure of the 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\} $.

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

The complement of $A$ is:

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

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

The closure of the complement is the smallest closed set that contains $\{b\}$, which is $\{b, c\}$.

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

Therefore, the boundary of $A$ is:

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

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

And that's how it's done.