MTH634 Assignment 1 Solution Fall 2026

This assignment explores fundamental concepts in topology, including open and closed sets, derived sets, closures, and comparisons between different topologies. Students will work with specific sets and collections on finite sets, the set of integers with the cofinite topology, and the real line with the usual topology. The tasks require analyzing properties of these structures and providing justifications for each conclusion. Students must produce a document containing their mathematical derivations and explanations.

1. 1

   Question 1

   5 marks

   Evaluate three given collections on X = {1, 2, 3, 4} to determine if they are topologies. For any collection that fails, identify the specific condition violated and provide an example. Additionally, determine the minimum number of subsets needed to make one of the collections a topology and define the resulting set.

2. 2

   Question 2

   5 marks

   Analyze the cofinite topology on the set of integers. Determine if four specific sets are open, providing reasons for each. Describe all closed subsets, evaluate the closure of two specific sets, and investigate the intersection of a sequence of open sets to discuss infinite intersections.

3. 3

   Question 3

   5 marks

   Given a specific topology on X = {a, b, c, d, e}, list all closed subsets. Calculate the derived set for a given subset, find its closure using two distinct methods, and determine if specific sets are closed or dense in X.

4. 4

   Question 4

   5 marks

   Work with three defined topologies on X = {1, 2, 3}. Perform intersection and union operations on these topologies, justifying whether the results are topologies themselves. Compare the topologies to determine if they are comparable and identify which is finer.

5. 5

   Question 5

   5 marks

   Examine the real line with the usual topology. Determine the openness of four specific sets with reasoning. Find the derived sets for three given subsets, identify which of these are closed, and calculate their closures.

What to submit

One MS-Word file containing the mathematical solutions.

Topics covered

Topologies Open and closed sets Cofinite topology Derived sets Closure of sets Comparable topologies Usual topology on R