Determine the set of limit points of the singlepoint set n i

Determine the set of limit points of the single-point set {n} in the digital line topology, justify your answer. (The result depends on whether n is even or odd.)

Solution

Given any open set n R,

Let x be a point in n. Since n is open, there will be a number >0 such that (x,x+) n.

We will now prove x is a limit point of n. For this, let >0 be given.

We can assume, that 0 < < . Now, (x,x+) (x,x+) n, therefore we can say (x,x+) n = (x,x+).

Since (x,x+) intersects n in a point other than x, we conclude x is a limit point.

=> all points x n are limit points.