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

To determine the boundary (∂A) of the set $ A = \{c\} $ in $ X = \{a, b, c\} $ with the topology $\{X, \{a\}, \{a, b\}, \emptyset\}$, we need to find the closure ($\text{Cl}(A)$) and the interior ($\text{Int}(A)$) of $A$, and then calculate the boundary as the difference between the closure and the interior.

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

The closed sets are thus: $\{X, \{c\}, \{b, c\}, \emptyset\}$.

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

In this case, the smallest closed set containing $\{c\}$ is $\{c\}$ itself.

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

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

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

There is no open set contained within $\{c\}$, as none of the given open sets include $c$. Therefore, the interior of $A$ is the empty set.

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

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

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

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

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

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

Alternative Method

Using the definition of the boundary as the intersection of the closure of $A$ and the closure of the complement of $A$:

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

The smallest closed set containing $\{c\}$ is $\{c\}$.

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

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

The closure of the complement of $A$ is the smallest closed set containing $\{a, b\}$.

In this topology, the closed sets are $\{X, \{c\}, \{b, c\}, \emptyset\}$.

The smallest closed set containing $\{a, b\}$ is $X$ itself, that is, $\{a, b, c\}$.

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

Now, we calculate the intersection of the closures:

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

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

The intersection of these sets is $\{c\}$.

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

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

And so on.