The plane minus $n$ points deformation retracts onto... I know that $\mathbb{R} ^ 2$ minus a point deformation retracts onto a circle
What I'm looking to prove is that $\mathbb{R}^2$ minus $n$ points deformation retracts onto the wedge sum of $n$ circles, without explicitly stating the deformation retraction
I can see it by drawing it, but I'm not sure how to go about proving it. I thought about proving that since $\mathbb{R} ^ 3 $ minus parallel lines is homeomorphic to $\mathbb{R}^2$ minus $n$ points and since  $\mathbb{R} ^ 3 $ minus parallel lines deformation retracts onto the wedge sum of $n$ circles then so does $\mathbb {R}^2$ minus $n$ points, but in this case I dont know how to prove that $\mathbb{R} ^ 3 $ minus parallel lines deformation retracts onto the wedge sum of $n$ circles
Any help would be much appreciated
 A: When you say "the wedge", I interpret this as "a wedge" i.e. a subset of the plane homeomorphic to the wedge of circles.
Since this is likely to be a homework, I will only give you a sequence of steps:

*

*If $X, Y$ are any two finite subsets of the same cardinality in  $R^2$, then there exists a homeomorphism $R^2\to R^2$ which carries $X$ to $Y$.

This, it does not matter which of the two finite subsets of the plane you have to consider. (This part of the problem was already solved at least once on MSE.)


*Suppose that $D$ is an open bounded convex subset of $R^2$. Then for every $x\in D$ there exists a deformation-retraction $D-\{x\}\to \partial D$.


*There exists a wedge of $n$ circles
$$
W= \vee_{i=1}^n C_i
$$
in $R^2$ such that each circle $C_i$ bounds a convex domain $D_i$ in $R^2$. Here, all the circles intersect at $0$. For each $i$ pick a point $x_i\in D_i$.


*Let $S^2= R^2\cup\{\infty\}$, the Riemann sphere (the 1-point compactification of $R^2$). Let $D_0$ denote the component of $S^2 - W$ containing the point $\infty$ and let $\bar{D}_0$ denote its closure. Show that there exists a continuous map of the closed disk $f: \bar{D}^2\to \bar{D}_0$ such that:
(a) $f: D^2\to D_0$ is a homeomorphism.
(b) $f$  is 1-1 overall except for $n$ boundary points of $D^2$ which are all mapped to $0\in \bar{D}_0$.
(In fact, you only need Part (a) but any reasonable construction will satisfy (b) as well.)
If you want to write a detailed proof, this is the hardest step.


*Use 4 to show the existence of a deformation retraction $D_0-\{\infty\}\to \partial D_0= W$. Use 2 to construct deformation retractions $D_i-\{x_i\}\to C_i$ for each $i=1,...,n$.


*Verify that all these deformation retraction taken together yield a deformation retraction
$$
R^2 - W\to R^2- X,
$$
where $X=\{x_1,...,x_n\}$.
