Metric and Topological Spaces

## Exercises 1

- Consider the upper-case letters of the alphabet { A, B, C, D, ... , Z } as being made up of (infinitely thin) lines. Classify them up to topological equivalence.

If we treat them as being made of lines of finite thickness (so that they are two-dimensional sets) how does the classification change?

If we treat them as being carved out of (say) wood (so that they are three-dimensional sets) does the classification change again?
Solution to question 1

- Recall the definitions:

A *closed interval *is the set [*a*, *b*] = {*x* **R** | *a* *x* *b* }

An *open interval* is the set (*a*, *b*) = {*x* **R** | *a* < *x* < *b* } and there are also open intervals

(*a*, ) = {*x* **R** | *a* < *x* }, (-, *b*) = {*x* **R** | *x* < *b* } and (-, ) = **R**.

A *half-open interval* is the set [*a*, *b*) = {*x* **R** | *a* *x* < *b* } or (*a*, *b*] = {*x* **R** | *a* < *x* *b* } and there are also half-open intervals [a, ) and (-, b] defined similarly.

Draw the graphs of continuous maps which show that any two closed intervals are homeomorphic (topologically equivalent).

Prove that any two *finite* open intervals are homeomorphic.

Prove that the open interval (-p/2, p/2) is homeomorphic to the real line **R**.

[Hint: Consider the map *f*(*x*) = tan(*x*).]

Prove that any two open intervals are homeomorphic.

Prove that any two half-open intervals are homeomorphic.
Solution to question 2

- Draw the graph of a continuous function from the open interval (0, 1) onto the closed interval [0, 1].

Can you find a continuous function from the interval [0, 1] onto the interval (0, 1) ?
Solution to question 3

- Show that the two embeddings of (the surface of the) two-holed torus shown on the right can be deformed into one another in
**R**^{3}.

[Hint: Think of the two-holed torus as being made from a sphere by attaching two "tubes".]

Show how the two views below of a ring and a two-holed torus can be deformed into one another in **R**^{3}.

Solution to question 4

SOLUTIONS TO WHOLE SET

JOC February 2004