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

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

The topology defines the following sets as open: $\{X, \{c\}, \{b, c\}, \emptyset\}$.

The closed sets are the complements of the open sets, so the closed sets are $\{X, \{c\}, \{b, c\}, \emptyset\}$.

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

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

First, let's determine the closure of $A$.

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

Since $\{b, c\}$ is already a closed set, we conclude that the closure of $A$ is $ \{b, c\} $ itself.

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

Next, we calculate the interior of $A$.

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

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

There are no open sets contained within $\{b, c\}$, as none of the given open sets include both $b$ and $c$ without also including $a$.

Therefore, the interior of $A$ is the empty set.

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

Now, we can determine the boundary of $A$ as the difference between its closure and its interior.

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

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

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

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

Alternative Method

We can also determine the boundary (∂A) of the set $A = \{b, c\}$ in $X = \{a, b, c\}$ with the topology $\{X, \{a\}, \{a, b\}, \emptyset\}$ using the definition:

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

We already know the closure of $A$ from the previous calculation:

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

Now, we need to determine the closure of the complement of $A$ ($\text{Cl}(X - A)$).

The complement of $A$ is:

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

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

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

Therefore, the smallest closed set that contains $\{a\}$ is $X$ itself, which is $\{a, b, c\}$.

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

Finally, we calculate the boundary of $A$ as the intersection of the closures:

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

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

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

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

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

And that's how we find the boundary.