Field Topology
Field Topology
[A row of four signs, each held up by two posts, followed by a row of four rounded lozenge shapes, one for each sign. The signs and lozenge shapes are shaded as if three-dimensional objects, all being flattish with a small third dimension; the four lozenge shapes each have one pair of sides horizontal and the other pair at a slight angle from vertical, denoting a horizontal plane perpendicular to the signs extending "out" towards the viewer, which places each shape "in front" of its sign. All but the first lozenge shape have various numbers of ellipses within the shape - ovoids shaded to denote holes piercing through the objects.]
[Leftmost sign:] Baseball Soccer Tetherball [The shape below this sign contains no ellipses.]
[Second sign from left:] Volleyball Badminton High jump [This shape has one large ellipse in the center.]
[Third sign:] Basketball Football Parallel bars [This shape has two large ellipses - one in the top half and one in the bottom half.]
[Fourth and rightmost sign:] Olympic swimming Croquet [This shape has nine small ellipses - eight arranged symmetrically towards the edges of the shape and one in the center.]
[Caption underneath the signs and shapes:] No one ever wants to use the topology department's athletic fields.

Field Topology is a subject in mathematics , but in this comic, Randall is instead examining the topology of playing fields used for various sports. The comic strip depicts a situation in which the common practice of multi-use athletic facilities has been organized by the "topology department" and constructed to be shared by all sports whose normal playing fields are topologically equivalent . One key assumption in topology is that you can ignore the specificities of shape, size and material of the objects concerned. This presents an amusing contrast as the "equivalent" topology department playing fields are actually not very appropriate for the activities listed in the comic, as the standard positioning, size and shape of hoops, nets and bars and the material of the field itself are not equivalent to the real playing fields used for those activities.
(Not to be confused with mathematical fields , or the Fields Medal prize -- although the concept is likely a further pun in the comic, as math (including topology), and most things once can imagine really, are mostly performed ("played") within mathematical fields.)
In topology, shapes which can be smoothly deformed into one another without adding or removing holes are considered equivalent. A topological hole is an area of the nominal space (or area, or other manifold) through which nothing restricted to this topology can pass. A loop is a path across the allowable territory of a topology (or a viable circuit to make through the world it describes) that end up where it started. For example, when describing the space taken up by a solid object such as a coffee mug, the handle forms a loop with a hole through it. If a loop cannot be tightened (ultimately adjusted to take a shorter path) down to a single point, then it must be wrapped around at least one "topological hole", and you have separately unique paths (or points, i.e., on different disconnected topologies) where you cannot adjust one loop to take the route of another without severing a looped path and reconnecting it.
When describing a negative space, such as the space around an archway, the 'hole' would be the material of the arch itself. This is because a loop formed by a ring around any part of the arch material can only be shortened to a finite length, not to a point; the 'hole' is the arch-shaped obstruction which forces the existence of these loops. A basketball hoop connected to the ground forms a similar obstruction with a loop through it, so the space around the hoop contains an equivalent hole. In this comic the topology department has analysed the spaces where various sports are played by the number of such obstructions in the playing area. Each space depicted in the comic is then signposted with the sports which are played on a field with that number of holes.
Baseball , tetherball and soccer are played on fields which are continuous in three-dimensional space. This means it is possible to traverse any path around or over any of the structures defining the field, while there are no obstructions which can be traversed through in a loop around them. The goals on a soccer field presumably do not create holes because the goalposts and crossbar are connected to the field by the net; Randall apparently considers these to form continuous surfaces which do not allow loops through them.
Volleyball and badminton are played using a net suspended from poles, and the high jump has a bar that contestants jump over. The structure formed by the net or bar and the supporting poles can be considered to be a "hole" through the playing field, as a path over and under the net/bar forming a loop cannot be contracted to a single point, so their playing fields in the comic all have one "hole".
A basketball court has two hoops. Parallel bars can be thought of as two archways. Both have opportunities to pass through either (or both) structures, and so the material of the structures define a hole in the topological abstract of the playing 'space'.  Since we are told that these sports fields belong to the Topology Department - and are not necessarily generalized to all sports fields - we might assume that their "football" field is either for rugby or for American football using H-shaped uprights .
An Olympic-sized swimming pool has ten lanes, and thus nine lane dividers which are fastened to the walls of the pool at each end, creating topological holes through the play area. Each hoop in croquet is similarly a hole through the space; while most versions of croquet use six hoops, nine hoops are used for "backyard croquet" which is played recreationally in the United States and Canada. The fact that the space in a swimming pool is typically filled with water [ citation needed ] has been overlooked by the topology department.
As mentioned in the title text, this last configuration is also homeomorphic to a foosball table (with each rod sustaining the player figures above the table defining a hole) or a Skee-Ball lane (which is even more straightforward, as it is just a plane with several holes in which to throw balls). These "fields" don't actually have the same number of holes, but are apparently lumped together by the Topology Department as having "many" holes.
Unfortunately, the Topology Department does not seem to have a field for hurdling events.
[A row of four signs, each held up by two posts, followed by a row of four rounded lozenge shapes, one for each sign. The signs and lozenge shapes are shaded as if three-dimensional objects, all being flattish with a small third dimension; the four lozenge shapes each have one pair of sides horizontal and the other pair at a slight angle from vertical, denoting a horizontal plane perpendicular to the signs extending "out" towards the viewer, which places each shape "in front" of its sign. All but the first lozenge shape have various numbers of ellipses within the shape - ovoids shaded to denote holes piercing through the objects.]
[Leftmost sign:] Baseball Soccer Tetherball [The shape below this sign contains no ellipses.]
[Second sign from left:] Volleyball Badminton High jump [This shape has one large ellipse in the center.]
[Third sign:] Basketball Football Parallel bars [This shape has two large ellipses - one in the top half and one in the bottom half.]
[Fourth and rightmost sign:] Olympic swimming Croquet [This shape has nine small ellipses - eight arranged symmetrically towards the edges of the shape and one in the center.]
[Caption underneath the signs and shapes:] No one ever wants to use the topology department's athletic fields.
