Exercise on Limit Points in Topology 3

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

The topology on \( X = \{a, b, c\} \) is \(\{X, \emptyset, \{a\}, \{a, b\}\}\). The open sets are:

- \( \emptyset \)
- \( \{a\} \)
- \( \{a, b\} \)
- \( X \) (i.e., \( \{a, b, c\} \))

To find the accumulation points of \( A = \{b\} \), I need to analyze each point \( x \) in the space \( X = \{a, b, c\} \) and identify those where every neighborhood contains at least one point of \( A \) other than \( x \).

• Point \( a \)
  - The neighborhoods of \( a \) are \(\{a\}\), \(\{a, b\}\), and \( X \).
  - \(\{a\}\) does not contain \( b \).
  - \(\{a, b\}\) contains \( b \).
  - \( X \) contains \( b \).
  Since not all neighborhoods of \( a \) contain \( b \), an element of \( A \) different from \( x = a \), \( a \) is not a limit point of \( A \).
• Point \( b \)
  The neighborhoods of \( b \) are \(\{a, b\}\) and \( X \).
  - \(\{a, b\}\) contains \( b \), but according to the definition of limit points, there must be at least one point of \( A \) different from \( x \), and here \( x = b \).
  - \( X \) contains \( b \), but again, it only includes the point \( b \) itself, which does not meet the requirement of a limit point, as we need a point different from \( x = b \).
  Therefore, \( b \) is not a limit point of \( A \) because no neighborhood of \( b \) contains another point of \( A \) different from \( b \).
• Point \( c \)
  The only neighborhood of \( c \) is \( X \).
  - \( X \) contains \( b \), which is in \( A \) and is different from \( x = c \). Therefore, it satisfies the condition.
  Every neighborhood of \( c \) contains \( b \), which means \( c \) is a limit point of \( A \).

In conclusion, only the point \( c \) is an accumulation point of the set \( A = \{b\} \) in the space \( X = \{a, b, c\} \) with the topology \(\{X, \emptyset, \{a\}, \{a, b\}\}\).

Therefore, the set of limit points of \( A \) is \(\{c\}\).

And that’s all there is to it.