Let X be a topological space and let Y be a compact space. Show that the...

 

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Let X be a topological space and let Y be a compact space. Show that the projection map ₁ X × Y → X is a closed map (images of closed sets are closed). (Hint: Show that the image of a closed set is closed directly, by showing it's complement is open. You will want to use the tube lemma as part of your proof.) Let X be a topological space and let Y be a compact space. Show that the projection map ₁ X × Y → X is a closed map (images of closed sets are closed). (Hint: Show that the image of a closed set is closed directly, by showing it's complement is open. You will want to use the tube lemma as part of your proof.)