The following is a result from Hatcher's Algebraic Topology text:
In various solutions to problems from the text that reference this corollary, it is stated that the space $X_G$ described above is path-connected. Why is this?
$X_G$ is certainly a CW complex, and it is known that a CW complex is path-connected if and only if it is connected. So, is there a relatively easy way to see that $X_G$ is connected, or a relatively easy way to see that $X_G$ is path-connected?
Thank you!
The wedge sum itself is clearly path connected, and attaching discs to it isn't going to stop it being path connected.
I guess if you want a formal proof, suppose you have a point. It's either the wedge point, or a point of one of the circles of the wedge sum, or a point on one of the discs. If it's a point on a disc, take a line from that point to the boundary of the disc and you get a path from your point to a point on the wedge sum. Then take a path along that circle to the wedge point, and all together you have a path from an arbitrary point of $X_G$ to the wedge point.
But then you can get a path between any two points by going from your starting point, to the wedge point, to your finish point.