According to the Invariance of Domain Theorem (Brouwer), we may assume without loss of generality that the hunt takes place in the Sahara desert.

• If the desert is a separable space, there is a sequence converging to any lion. Choose one such and approach stealthily along it, bearing suitable equipment. If the desert is not separable, there is a net converging to the lion. Use it.

• Topologize the desert with the leonine topology, in which a set is closed if and only if it contains no lions, or is the whole desert. Then the set of lions is dense. Place an open cage in the desert, and shut the door quickly.

• A lion has at least the connectivity of a torus. Embed the desert in 4-space, and deform it so as to knot the lion. This renders it helpless, and essentially captured.

• A lion is a topological manifold with boundary. Any such can be collared (Brown).

• Regard a lion as an orientable 3-manifold. It can be made contractible via a sequence of surgeries. Contract the lion to a circus.

• A lion is a cross-section of the sheaf of germs of lions on the desert. Changing the topology of the desert to the discrete topology, the stalks decompose into their component germs and infect the lion. This makes the lion ill, hence helpless.

• Lions are hairy, so we regard them as made up of fibers, hence as fibrations. Construct a Moore-Postnikov decomposition. A decomposed lion is dead, in need of burial.

• A lion has the homotopy type of a 1-dimensional complex, hence is a K(G,1). If G is not abelian, then the lion is not a member of the International Commutist Conspiracy, thus friendly. If G is abelian, then the lion has the homotopy type of the loop space of K(G,2). Hire a stunt pilot to loop the loops, thereby thoroughly entangling the lion.