Find a homeomorphism

Question

Let $X=A\cup B \cup C$ where $A=\{(x,y) :(x+2)^2 +y^2 =1\}$ and $B=\{x^2+y^2 \leq 1\}$ and $C=\{(x,y) :(x-2)^2 +y^2 =1\}$.

Find a homeomorphism between the quotient space $X/B$ and $E=\{(x,y) :(x-1)^2 +y^2 =1\} \cup \{(x,y) :(x+1)^2 +y^2 =1\}$

Answer 1

Hint: define a continuous surjection $f : X \to E$. For this, make use of the gluing lemma: the function will be continuous if and only if it is continuous restricted to $A$, $B$ and $C$, as they are closed and their union is the whole of $X$.

Now, show that this map is $B$-compatible, i.e. that if $x,y \in B$ then $f(x) = f(y)$. This proves that $f$ will factor through the quotient, or in other words, that we have a continuous mapping $g : X/B \to E$ defined as $g(x) = [f(x)] \in X/B$.

Finally, prove that $g$ is injective (and thus bijective, as surjectivity comes from the previous factorization) which immediately proves that $g$ is a homeo, as $X/B$ is compact (it is the image via the projection of $X$ which is compact), and $E$ is Hausdorff.