I'm reading this blog post where they state that $\Bbb{R}^3 - S^1$ deformation retracts to $S^2 \vee S^1$ and then they proceed to give the following sketch
with the explanation
You can see this in the drawing above where $X = \Bbb R^3 \setminus \{ \text{circle} \}$ consists of the space inside the yellow cube including the blue disc, but not including the white boundary of that disc. (That white part is our deleted circle which I will from now on refer to as "our/the circle.") Notice that our circle sits on some plane in $\Bbb R^3$. (I've shaded it in dark yellow.) Then all the points lying on that plane outside of the circle (the dark yellow bit) along with the points in $\Bbb R^3$ not on the plane (the light yellow bit) deformation retract onto a sphere. (So basically, everything that's yellow maps to a sphere.) And all the points enclosed by the circle (the blue disc) deformation retract onto a line, i.e. the diameter of our sphere.
The white circle is not very clear from the picture, but it's the boundary of the blue disc in the first picture if someone misses it.
I usually understand the process, but here the first $\simeq$ feel very odd. Where does the white circle go? Also are we including the inner part of the "sphere" in this deformation or is it just the boundary? I know that $\Bbb R^3$ can be deformed to the three dimensional disc $D^3$ and $\Bbb R^3 - \{ \text{point} \}$ to $S^2$.
It's added on the blog later that
Added May 22, 2016: I realize (Thank you, readers!) that the description of the first homotopy equivalence in the picture above is a bit cryptic. And to make matters worse, that horizontal blue line should really be vertical! Yikes. So instead of trying to illuminate (and thus possibly obscure) the situation with more words, let me show you a drawing of what's really going on (at least, in my brain). You'll have to excuse my shabby artistic skills, but hopefully the deformation retract is a little clearer now:
but this one seems even more confusing. It's like there is no circle at all inside this solid sphere in this sketch?
I agree that the first sketch is probably not worth spending a lot of time deciphering.
In the first drawing of the second sketch, you have to imagine the white space around the "bulb" in the middle really goes all the way around the bulb, and that the bulb connects to the "outer shell" that is the sphere at its narrow top and bottom. That white space is the missing circle.