MT560 Algebraic Topology Autumn 2016 – Homework 2


MT560 Algebraic Topology Autumn 2016 – Homework 2
1. Suppose that X,Y are spaces with Y Hausdorff and f : X → Y is continuous. Show that the graph of f, Gr(f) = {(x,f(x)) ; x ∈ X}, is a closed subset of X ×Y .
2. Let (X,T) be a topological space and ‘∗’ a ‘point’ not in the set X. Let X0 = X ∪{∗}. (a) Let T0 = T ∪{X0}. Show that T0 is a topology on X0. (b) Show that X is a subspace of X0. (c) Show that (X0,T0) is compact. (d) Suppose X is Hausdorff. Does it follow that X0 is Hausdorff?
3. Alexandroff 1-point compactification. Let (X,T) be a topological space and ∗ a ‘point’ not in the set X. Let ˆ X = X ∪{∗}. (a) Let TA := T ∪{U ∪{∗} ; X − U is closed and compact in X}. Show that ( ˆ X,TA) is a compact topological space containing X asa subspace. (Note that X is open in ˆ X.) (b) Suppose that X is Hausdorff and locally compact (i.e. for each x ∈ X, there exists an open set U such that x ∈ U, and its closure, ¯ U, is compact). Deduce that ˆ X is Hausdorff. (c) Suppose X is not compact. Deduce that X is dense in ˆ X. Is this the case if X is compact? (d) Show that the Alexandroff 1-point compactification of the open n- cell en is homeomorphic to the n-sphere Sn. Show that attaching the n-disc Dn to a 0-cell ∗, via the constant map f : ∂Dn →{∗}, also gives the n-sphere.

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