3 Limit points

Question 1

Limit point (accumulation point, cluster point)

Answer 1

Let A be a subset of a topological space (X, T). A point x e X is said to be a limit point or accumulation point or cluster point) of A if every open set, U, containing x contains a point of A different from x.

Question 2

Closure

Answer 2

Let A be a subset of a topological space (X, T), and A’ the set of all limit points of A. Then the set A u A’ consisting of A and all its limit points is called the closure of A and is denoted by A^-.

Question 3

Dense (everywhere dense)

Answer 3

Let A be a subset of a topological space (X, T). Then A is said to dense in X or everywhere dense in X if A^- = X.

Question 4

Neighbourhood

Answer 4

Let (X, T) be a topological space, N a subset of X and p a point in N. Then N is said to be a neighbourhood of the point p if there exists an open set U such that p e U (_ N.

Question 5

Connectedness

Answer 5

Let (X, T) be a topological space. Then it is said to be connected if the only clopen subsets of X are X and O/.