By J. P. May
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Extra info for A Concise Course in Algebraic Topology
It is a categorical fact that functors which are right adjoints preserve limits, so this does give categorical limits in U . This is how we defined X × Y , for example. Point-set level colimits of weak Hausdorff spaces need not be weak Hausdorff. However, if a point-set level colimit of compactly generated spaces is weak Hausdorff, then it is a k-space and therefore compactly generated. We shall only be interested in colimits in those cases where this holds. The propositions above give examples.
2) * A Tychonoff (or completely regular) space X is a T1 -space (points are closed) such that for each point x ∈ X and each closed subset A such that x∈ / A, there is a function f : X −→ I such that f (x) = 0 and f (a) = 1 if a ∈ A. , Kelley, General Topology). (a) A space is Tychonoff if and only if it can be embedded in a cube (a product of copies of I). (b) There are Tychonoff spaces that are not k-spaces, but every cube is a compact Hausdorff space. (3) Brief essay: In view of Problems 1 and 2, what should we mean by a “subspace” of a compactly generated space.
For each subgroup H of G, the covering p : E(G/H) −→ B has a canonical basepoint e in its fiber over b such that p∗ (π1 (E(G/H), e)) = H. ∼ Moreover, Fb = G/H as a G-set and, for a G-map α : G/H −→ G/K in O(G), the restriction of E(α) : E(G/H) −→ E(G/K) to fibers over b coincides with α. Proof. Let p : E −→ B be the universal cover of B and fix e ∈ E such that p(e) = b. We have the isomorphism Aut(E) ∼ = π1 (B, b) given by mapping g : E −→ E to the path class [f ] ∈ G such that g(e) = T (f )(e), where T (f )(e) is the endpoint of the path f˜ that starts at e and lifts f .