Is there a continuous injection from the unit square to the unit interval?

I see that the Peano curve is a continuous surjection from the unit interval to the unit square (correct me if I'm wrong). Does it then follow that there is a continuous injection from the unit square to the unit interval?

Thank you!

No. The image of the square under a continuous function to the real line is a compact interval. Removing the center of the square leaves it connected. Removing an interior point from an interval disconnects it.

Suppose that $f:[0,1]^2\to[0,1]$ is a continuous injection. $[0,1]^2$ is compact and connected, so $f\big[[0,1]^2\big]$ must be compact and connected; since it’s not a singleton, it must be a closed interval, and without loss of generality we may assume that it is $[0,1]$ itself, so that $f$ is a continuous bijection.

Now observe that $[0,1]\setminus\left\{\frac12\right\}$ is not connected, but $[0,1]^2\setminus f^{-1}\left(\frac12\right)$ is connected, and therefore $f$ is not continuous after all.