***JUST NUMBER (3) PLEASE! THANKS!***

Assume all spaces are T3 (1) Use the Tube Lemma (from, and adapt, our proof that compact x compact is compact) to prove that if X is compact and Y is Lindelf, then X X Y is Lindelf. (2) If X is sequentially compact and Y is countably compact then X X Y is countably compact. Hint: First step is like the proof that the product of two sequentially compact spaces is sequentially compact. (3) A subset A of a space X is said to be C-embedded in X if every continuous g: A + R has a continuous extension f :X + R. (a) Prove that if A is a compact subspace of Tychonoff space X then A is C-embedded in X. Hint: you can use that X has a compactification. (b) Describe (and prove claims) an example to show that a closed Lindelof subspace of a Tychonoff space X may fail to be C-embedded. Hint: Tietze's extension Lemma implies there is no normal example. Assume all spaces are T3 (1) Use the Tube Lemma (from, and adapt, our proof that compact x compact is compact) to prove that if X is compact and Y is Lindelf, then X X Y is Lindelf. (2) If X is sequentially compact and Y is countably compact then X X Y is countably compact. Hint: First step is like the proof that the product of two sequentially compact spaces is sequentially compact. (3) A subset A of a space X is said to be C-embedded in X if every continuous g: A + R has a continuous extension f :X + R. (a) Prove that if A is a compact subspace of Tychonoff space X then A is C-embedded in X. Hint: you can use that X has a compactification. (b) Describe (and prove claims) an example to show that a closed Lindelof subspace of a Tychonoff space X may fail to be C-embedded. Hint: Tietze's extension Lemma implies there is no normal example