Proving two spaces homeomorphic means producing a map. Proving they are not means finding a property that one has and the other lacks, and that any homeomorphism would have to carry across. Such a property is called a topological invariant, and collecting them is how the rest of the course is used.

To show two keys are different it is enough to find one lock that one opens and the other does not. The locks are the invariants, and the rest of the course is a locksmith.

A property of spaces is a topological invariant when $X\cong Y$ and $X$ having it force $Y$ to have it. Every property stated purely in terms of open sets is an invariant, because a homeomorphism matches the topologies exactly; the invariants worth having are those that can be computed or denied in examples.

• The number of points, and the number of open sets: a homeomorphism is a bijection on points and induces one on topologies.
• Being discrete, being indiscrete, having an isolated point: all of them are statements about open sets alone.
• Being metrisable — that is, having a topology that comes from some metric — is an invariant, and it is what separates the Sierpiński space from every metric space.
• Boundedness, the length of an interval and completeness are not invariants: $(0,1)\cong\mathbb{R}$ settles all three at once.
• The invariants this course is really built for arrive one chapter at a time from here on. Each of them is defined in open sets alone, so each is automatically an invariant, and each is met as the answer to a question this chapter cannot settle.

The map is open, because its inverse is continuous, so an open set of the source has an open image. The map is continuous, so an open set of the target has an open preimage; and for a bijection the image and the preimage undo each other. The two directions give a bijection between the two topologies, and any property counted or stated in open sets is therefore the same on both sides.

The map is open, because the inverse is continuous.

The map is continuous, so preimages of open sets are open.

For a bijection the two operations undo each other, so the correspondence is one to one.

The topologies are matched, so any property written in open sets alone holds on one side exactly when it holds on the other.

An invariant is a property that a homeomorphism cannot destroy. Properties written in open sets alone are automatically invariant.

• An invariant is a property homeomorphic spaces share.
• Anything stated in open sets alone is automatically invariant.
• The number of points, the number of open sets, discreteness and metrisability are invariants.
• Boundedness, length and completeness are not.

Great work! You now know how to:

• say what a topological invariant is;
• use one to show two spaces are not homeomorphic;
• name invariants available so far and two properties that are not;
• explain why $(0,1)\cong\mathbb{R}$ settles boundedness, length and completeness at once.