1) What does the subset XY=UV of Euclidean space of dimension 4 look like?

2) How many disjoint copies of a figure eight "8" can be embedded in the plane, countably or uncountably many? What about if "8" is replaced by "Y" ?

3) A planar surface is an open subset of the plane. Are there uncountably many connected planar surfaces no two of which are homeomorphic? Are there two connected planar surfaces without boundary having the same fundamental group but which are not homeomorphic?.

4) Is it possible to have two topological spaces X and Y which are not homeomorphic but so that each space is (homeomorphic to) a finite sheeted covering space of the other?

5) A table has 3 legs of equal length. Is it always possible to place the table on a convex hill so that the surface is level?

6) There are equal numbers of black and white points in the plane. No 3 points lie in a straight line. Is it always possible to draw straight lines so that each straight line starts on a white point and ends on a black point, so that the lines do not meet, and so that every point is on exactly one line?.

7) A finite graph is a finite set of points in space called vertices together with a finite number of edges connecting pairs of vertices. Each edge meets exactly 2 vertices, one at each of its endpoints. If one edge meets another edge they meet only at one endpoint. Prove that if a graph has more than 1 vertex, then there are 2 vertices with the same degree (=the number of edges meeting the vertex)

8) If you take a square and look at it from some point in space it looks like a quadrilateral. What are the possible shapes of this quadrilateral?

9) Is there a subset of Euclidean 3-space with an element of finite order (not the identity) in its fundamental group?