By J. P. May
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Extra resources for A Concise Course in Algebraic Topology
A subspace A of X is said to be “compactly closed” if g −1 (A) is closed in K for any map g : K −→ X from a compact space K into X. When X is weak Hausdorff, this holds if and only if the intersection of A with each compact subset of X is closed. A space X is a “k-space” if every compactly closed subspace is closed. A space X is “compactly generated” if it is a weak Hausdorff k-space. For example, any locally compact space and any weak Hausdorff space that satisfies the first axiom of countability (every point has a countable neighborhood basis) is compactly generated.
8. THE CONSTRUCTION OF COVERINGS OF SPACES 33 6. Let X be a G-space. Show that passage to fixed point spaces, G/H −→ X H , is the object function of a contravariant functor X (−) : O(G) −→ U . CHAPTER 4 Graphs We define graphs, describe their homotopy types, and use them to show that a subgroup of a free group is free and that any group is the fundamental group of some space. 1. The definition of graphs We give the definition in a form that will later make it clear that a graph is exactly a one-dimensional CW complex.
Then passage to orbit spaces defines a functor X/(−) : O(G) −→ U . Proof. The functor sends G/H to X/H and sends a map α : G/H −→ G/K to the map X/H −→ X/K that sends the coset Hx to the coset Kγ −1 x, where α is given by the subconjugacy relation γ −1 Hγ ⊂ K. The starting point of the construction of general covers is the following description of regular covers and in particular of the universal cover. Proposition. Let p : E −→ B be a cover such that Aut(E) acts transitively on Fb . Then the cover p is regular and E/ Aut(E) is homeomorphic to B.