The complement of an algebraic set in is path-connected.

# topology

# Contractible manifolds

Classify the closed contractible manifolds (Recall that a manifold is closed if it is compact and its boundary is empty.)

# Connect the dots

Construct a countable, connected topological space with at least two points, satisfying the highest separability axiom you can.

Notice that if two points can be separated by a continuous function, then a connected space with more than one point is uncountable, so your space cannot be (or higher).

# No swapping

There is no continuous function such that and .

# Fixed points of the ball

Brouwer’s theorem shows that any map of the unit ball to itself has a fixed point. Evidently the collection of fixed points is a closed subset of the ball. What closed subsets are realized as the fixed point set of a continuous map of the ball into itself?

# Complements of closed sets

Construct two homeomorphic closed subsets of some Euclidean space such that their complements are not homeomorphic, but show that if are closed homeomorphic subsets of such that lies in and lies in , their complements in are homeomorphic.

# Venn diagrams

Venn diagrams are useful since they represent every possible intersection between the sets that one is studying. Suppose that we want to draw a Venn diagram for a single set. Then it is enough to draw it in zero dimensions (the zero-dimensional space is just the space with one point): the set is represented by the single point in the zero-dimensional space. If we want to draw a Venn diagram that represents two sets we notice that the zero-dimensional space won’t do, there is not enough space. But in the one-dimensional space (the real line) we can draw both sets and in such a way that we have distinct regions for , and . Once again if we want to draw one more set, one dimension won’t do. But we can draw the classical Venn diagram in two dimensions.

*First exercise:* Show that we cannot draw a Venn diagram in two dimension for four sets using only circles.

*Second exercise:* Show that this can be done if we use for example rectangles.

*Third exercise*: Can we draw a Venn diagram for four sets in two dimensions in such a way that the region that represents every intersection is connected? And simply connected?