8 Topology Questions : Hausdorff Space, Countability, Compactness and Homomorphisms

1. Show that the collection
13 {[O,c)J 0<c 1}u{(d,1]|O<d< 1)
cannot be a base for the subspace topology [O, 1], where is the Euclidean topology on rt
Hint: Use contradiction, i.e., first assume that B is a base for [O, 1].
2. Let B {[a, b) x [c, d)Ia < b, : < d}. Show that B is a base for the product space (R x R, £ x £), where £ is the lower limit topology on 1?.
3. Let X. Y be spaces and f : X ?> Y a continuous surjcction. Prove that if X is P countable, so is Y.
4. Let X, Y he spaces and f : X ?> Y a continuous surjection. Prove that if X is compact, so is V.
3. Show that the union of finitely many compact subsets of a space is compact.
6. Let X be a compact space and V a Hausdorff space. Then show that f X ?> Y is a homomorphism if and only if f is bijeetive and continuous.
7. Let X be a Hausdorff space and P, Q disjoint compact subsets of X. Then show that there exist open sets U and V such that P C U, Q C V and U U V 0.
This property tells us that compact subsets of a Hausdorif space behaves like points.
8. Let X, Y be spaces arid f X ?> V a continuous surjection. Prove that if X is connected, so is V.