MAT246Y – Spring 2002 – Outline of Point Set Topology

(Handout prepared and distributed by P.Rosenthal)

1. In the real line, a set is open if it is a union of open intervals. In the plane, a set is open if it is the union of open disks.

2. A topological space is a set together with a collection of subsets, called the open sets, such that the union of any number of open sets is open, the intersection of a finite number of open sets is open, and the empty set and the entire space are both open.

3. Definition: In a topological space, a point x is a limit point of a subset S if every open set containing x intersects S in a point other than x ( x may be in S but need not be).

4. Definition: a set is closed if its complement is open.

Theorem: A set is closed if and only if it contains all its limit points.

5. Definition: the closure of a set is the union of the set and its set of limit points.

Problem #1: Prove that the closure of a set is always closed.

Problem #2: Prove that the closure of the union of two sets is the union of their closures.

6. Definition: a function from one topological space to another is continuous if the inverse image of every open set is open.

Problem #3: Prove that a function is continuous if and only if the inverse image of every closed set is closed.

Problem #4: (hard – feel proud if you completely solve this)

What is the maximum number of distinct subsets of the real line that can be obtained by starting with a given subset and successively applying closure and complement (in any order)?

7. Definition: A homeomorphism is a one-to-one function from a topological space onto another topological space which is continuous and which has a continuous inverse. Two topological spaces are said to be homeomorphic if there is a homemorphism from one onto the other.

8. Definition: a topological space is said to be disconnected if the space is the union of two closed disjoint non-empty subsets. A topological space is connected if it is not disconnected.

9. Definition: a topological property is a property that a topological space may have that is preserved under homeomorphism.

10. Theorem: The continuous image of a connected set is connected.

Corollary: Connectedness is a topological property.

11. Definition: If S is a subset of a topological space, then the relative topology on S is the topology in which a subset of S is open if and only if it is the intersection of S and an open subset of the entire space.

12. Theorem: A subset of the real line with the usual topology is connected if and only if it contains the interval [a,b] whenever it contains a and b.

Corollary: The intermediate value theorem.

13. Definition: A component of a space is a connected subset of the space which is not properly contained in any other connected subset of the space.

14. Theorem: If a collection of connected sets has the property that one given member of the collection intersects every other member then the union of the collection is connected.

Corollary: Every topological space is a disjoint union of its components.

15. Problem #5: Prove that a component of a topological space is a closed subset of the space.

16. Definition: A cut point of a connected set is a point such that the set minus that point is disconnected.

Theorem: The number of cut points that a connected space has is a topological invariant.

Corollary: The real line is not homeomorphic to the plane (with the usual topologies).

Problem #6: Prove that there is a topology on the plane under which it is homeomorphic to the line with its usual topology.

17. Problem #7: Prove that the number of components of a topological space is a topological invariant.

18. Problem #8: Determine which capital letters are homeomorphic to each other (when regarded as subspaces of the plane with the usual topology).

19. A subset of a topological space is said to be compact if every covering of the subspace by a collection of open sets has a finite sub-covering.

20. Heine-Borel Theorem: A subspace of the real line is compact in the usual topology if and only if it is closed and bounded.

21. Theorem: A continuous image of a compact space is compact.

Corollary: A continuous real-valued function on a closed interval has a maximum and a minimum value.

22. A closed subset of a compact space is compact.

Problem #9: Prove that a subset of the real line is compact and connected in the usual topology if and only if it is a closed interval.