Exercise on Accumulation Points in Topology 1

In this exercise, I need to determine the accumulation points (limit points) of the set A={a} within the topological space X={a,b,c}, given the topology {X,  Ø, {a}, {a,b}}.

The set to be analyzed consists of a single point: {a}

$$ A = \{ a \} $$

The open sets defined by this topology are as follows:

$$ \{ \ X , \emptyset, \{ a \} , \{a,b \} \ \} $$

Where X={a,b,c} represents the entire topological space that includes the set A.

$$ \{ \ \{a,b,c \} , \emptyset, \{ a \} , \{a,b \} \ \} $$

To find the accumulation points of A in the space X, I need to identify the points x in X such that every neighborhood of x contains at least one point of A different from x.

Let's examine each point in the space X={a,b,c}:

• x={a}
  The point x={a} is in the open sets {a,b,c}, {a}, and {a,b}. All neighborhoods of the point {a} contain at least one element of A={a}, but none different from x={a}. Therefore, the point x={a} is not an accumulation point of A.
• x={b}
  The point x={b} is in the open sets {a,b,c} and {a,b}. All neighborhoods of the point {b} contain at least one element of A={a} different from x={b}. Therefore, the point x={b} is an accumulation point of A.
• x={c}
  The point x={c} is only in the open set {a,b,c}. Even in this case, all neighborhoods of the point {c} contain at least one element of A={a} different from x={c}. Therefore, the point x={c} is an accumulation point of A.

In conclusion, the points {b} and {c} are accumulation points of the set A={a} in the space X={a,b,c} with the specified topology.

And so on.