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

In this exercise, I need to determine the boundary ($\partial A$) of the set $A = \{b\}$ 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, which are the complements of the open sets relative to $X$, are $\{X, \{b,c\}, \{c\}, \emptyset\}$.

The boundary of a set $A$ is defined as the difference between its closure and its interior:

$$ \partial A = Cl(A) - Int(A) $$

The set $A = \{b\}$ is a singleton set.

The closure of $A$ is the smallest closed set that contains $A$.

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

Thus, the smallest closed set containing $\{b\}$ is $ \{b, c\} $.

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

The interior of $A$ is the union of all open sets contained within $A = \{b\}$.

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

No open set is contained within $A = \{b\}$, so the interior of $\{b\}$ is the empty set.

$$ Int(A) = \emptyset $$

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

$$ \partial A = Cl(A) - Int(A) $$

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

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

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

Alternative Method

The boundary of a set can also be found by taking the intersection of the closure of the set $A$ and the closure of its complement $X-A$:

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

The closure of $A = \{b\}$ is:

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

The complement set $X - A$ is $ \{a, c\} $:

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

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

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

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

Therefore, the smallest closed set containing $\{a, c\}$ is $X = \{a, b, c\}$.

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

Now I can find the boundary as the intersection of the two closures:

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

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

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

The final result is the same.

And so on.