Cfrgtkky

Munkres Topology Example 70.1

Let $theta$ be theta-space, $theta_a := theta setminus {a}$ and $theta_b := theta setminus {b}$. Let $theta_{ab} := theta_a cap theta_b = theta setminus {a,b}$ be doubly punctured theta-space where $a,b$ are interior points of $A$ and $B$.

My guess is yes, $theta_a$ is contractible, and my intuition is that the addition of $b$ to $theta_{ab}$ for contractible $theta_{ab}$ is still contractible because $theta_a$ isn't any less path connected than $theta_{ab}$ if you think of a contractible space as shrinking to a point.

Here is what would be a proof:

We want to find continuous $f: theta_a to {0}$ and $g: {0} to theta_a$ such that $g circ f$ is homotopic to $id(theta_a)$ where $g(f(z))=g(0)$ and $f circ g = id({0})$.

Because $theta_{ab}$ is contractible, there are continuous $l: theta_{ab} to {0}$ and $m: {0} to theta_{ab}$ such that $m circ l$ is homotopic to $id(theta_{ab})$ where $m(l(z))=m(0)$ and $l circ m = id({0})$.

I think the inclusion function always exists and is continuous $j: theta_{ab} to theta_a$. If there exists a retraction $r: theta_{a} to theta_{ab}$, which is a continuous function such that $r circ j = id(theta_{ab})$, then we our $f$ and $g$ are $l circ r$ and $j circ m$, respectively because:

$g circ f = j circ m circ l circ r simeq j circ id(theta_{ab}) circ r = j circ r = id(theta_a)$

and

$f circ g = l circ r circ j circ m = l circ id(theta_{ab}) circ m = l circ m = id({0})$.

1. So it all comes down to whether or not there's a retraction. What's the rule? I know retracts of contractible are contractible but if we have a contractible space X that is a retract of Y, then $Y$ is contractible?




2. I'm not sure I use my path connected intuition here, but maybe there is a link between path connected and retract?




3. Also, if $X subset Y$ and $X$ and $Y$ are contractible, then is $X$ a retract of $Y$?




4. And finally, is $theta_a$ contractible by this route or some other route?

I think no retraction exists because $theta_{ab}$ is open in $theta_a$, ${b}$ is closed in $theta_a$ and $theta_a$ is connected because $theta_a$ is path connected, so there's no hope of a clopen set. I think $theta_a$ is contractible but instead for a similar reason as to why $theta_{ab}$ is contractible.