This picture is from the Encyclopedia of Mathematics's page on the Brouwer degree:
It might be necessary to right-click on the image and open it in a new tab so that it is a legible size. The reference mentioned is:
E. Zeidler, "Nonlinear functional analysis and its applications" , I , "Fixed Point Theorems", Springer (1986)
As you might have noticed, the math formatting on the website is somewhat poor, so I don't even understand what the statement of the property is supposed to be, thus I don't know where to look for it in Zeidler's book.
If anyone understands what the article is supposed to be saying, or if you even have a copy of Zeidler's book and know where this is referenced, I would be very grateful for your help.