The Basic Idea of Topology

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Section The Basic Idea of Topology

If you like geometry, you will probably like topology. Geometry is the study of objects with certain attributes (e.g., shape and size), while topology is more general than geometry. In topology, we aren’t concerned about the attributes (shape and size) of an object, only about those characteristics that don’t change when we transform the object in different ways (any way that doesn’t involve tearing or poking holes the object). There are lots of really interesting theorems in topology — for example, the Hairy Ball Theorem which states that if you have a ball with hair all over it (think of a tribble from Star Trek — if that isn’t too old of a reference), it is impossible to comb the hairs continuously and have all the hairs lay flat. Some hair must be sticking straight up!

Activity 1.2.

(a)

Take a pipe cleaner, a rubber band, or a piece of string and make a square from it. You are allowed to change the square by moving parts of the square without breaking it or lifting it off the surface it is on. To which of the following shapes can you transform your square? Explain.

(i)

a circle

(ii)

the letter S

(iii)

a five point star ☆

(iv)

the letter D

(b)

Now take some play-doh (if you don’t have any play-doh, just use your imagination). Use the play-doh (or your imagination) to determine which of the following shape can be transformed into others without breaking or making holes.

(i)

a filled square

(ii)

a doughnut

(iii)

a bowl

(iv)

a coffee mug with handle

This idea of transforming one set into another as we explored in Activity 1.2 is formally done with functions. As we progress through this subject, we will need to have more rigorous definitions of functions and sets. We begin with sets and discuss functions in Chapter 2.

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