MTH634 Quiz by Khan

Mth634 (Topology) Assignment #1-2023 · Solved By Pin✌️ And Virtualain 🕊️ · Solution:

Q1. As X= {1,2,3,4,5} is a discrete topology.

Now, we want to show that if a specific point 2 is a limit point of X={1,2,3,4,5} with the discrete topology.

As we know that in discrete topology, every point is an isolated point, which means that for each point X in topological space, there exists an open set containing X and no other points.

For the set X={1,2,3,4,5}, consider that the 2 is limit point.

A point is limit point if a set is every open set containing that point intersect the set at a point other than itself.

If intersect the other points of X in a open set then there is no other point in open set.

Therefore, 2 is not a limit point of X with the discrete topology because there is no open set containing 2 that intersect X at any other points.

Q2. Answer:

We want to show that these topologies are open. So;

By using the definition of co-finite topology, Set A is open because its complement contains only 2 elements which are finite.

Similarly;

By using the definition of co-finite topology, Set B is open because its complement contains only 3 elements which are finite.

Now in co-finite topology, sets are open and their complements are finite, so the set A and set B satisfied these requirements.

Therefore they are open set in the co-finite topology on X.