S38 - 2nd April 2005 Munkres 38 Ex 38.4 Let X → βX be...

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Unformatted text preview: 2nd April 2005 Munkres § 38 Ex. 38.4. Let X → βX be the Stone– ˇ Cech compactification and X → cX an arbitrary com- pactification of the completely regular space X . By the universal property of the Stone– ˇ Cech compactification, the map X → cX extends uniquely X / ! B B B B B B B B cX βX = z z z z z z z z to a continuous map βX → cX . Any continuous map of a compact space to a Hausdorff space is closed. In particular, βX → cX is closed. It is also surjective for it has a dense image since X → cX has a dense image. Thus βX → cX is a closed quotient map. T Ex. 38.5. (a) . For any ε > 0 there exists an α ∈ S Ω = [0 , Ω) such that | f ( β )- f ( α ) | < ε for all β > α . For if no such element existed we could find an increasing sequence of elements γ n ∈ (0 , Ω) such that | f ( γ n )- f ( γ n- 1 ) | ≥ ε for all n . But any increasing sequence in (0 , Ω) converges to its least upper bound whereas the image sequence f ( γ n ) ∈ R does not converge; this contradicts continuity of the function f : (0 , Ω) → R . So in particular, there exist elements α n such that | f ( β )- f ( α n ) | < 1 /n for all β > α n . Let α be an upper bound for these elements. Then f is constant on ( α, Ω). (b) . Since any real function on (0 , Ω) is eventually constant, any real function, in particular any bounded real function, on (0 , Ω) extends to the one-point-compactification (0 , Ω]. But the Stone- ˇ Cech compactification is characterized by this property [Thm 38.5] so (0 , Ω] = β (0 , Ω). (c) . Use that any compactification of (0 , Ω) is a quotient of (0 , Ω] [Ex 38.4]....
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S38 - 2nd April 2005 Munkres 38 Ex 38.4 Let X → βX be...

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