I'm trying to solve the following question:
Let $X$ be a topological space and $A$ a subset of $X$. On $X\times\{0,1\}$ define the partition composed of the pairs $\{(a,0),(a,1)\}$ for $a\in A$, and of the singletons $\{(x,i)\}$ if $x\in X\setminus A$ and $i \in \{0,1\}$.
Let $R$ be the equivalence relation defined by this partition, let $Y$ be the quotient space $[X \times \{0,1\}]/R$ and let $p:X \times \{0,1\} \to Y$ be the quotient map.
(1) (a) Prove that there exists a continuous map $f:Y \to X$ such that $f \circ p(x,i)=x$ for every $x\in X$ and $i \in \{0,1\}$.
(b) Prove that Y is Hausdorff if and only if X is Hausdorff and A a closed subset of X.
(2) Consider the above construction for $X = [0, 1]$ and A an arbitrary subset of $[0, 1]$. Prove that Y is compact. Prove that $K = p(X × \{0\})$ and $L = p(X × \{1\})$ are compact, and that $K ∩ L$ is homeomorphic to A.
I've managed to solve everything but the very last part, where I'm asked to show that $K ∩ L$ is homeomorphic to A. Clearly $K ∩ L = \{\{(a,0),(a,1)\} : a \in A\}$, and it's easy to find a bijection between the two (just take $f(\{(a,0),(a,1)\}) = a$), but I'm struggling to show this (and its inverse) is continuous.
I'd really appreciate any help you could offer.
The restricted map $p|X\times\{0\}\to K$ is a continuous bijection and we can straightforwardly check that it is open. So it is a homeomorphism. Then its restriction to a set $A\times\{0\}$ is a homeomorphism onto the image which is $K\cap L$. It remains to note that the space $A\times\{0\}$ is naturally homeomorphic to $A$.