The US states of Colorado, Utah, Arizona and New Mexico all meet at a point called the Four Corners.

The east-west boundary line between Colorado/Utah and Arizona/New Mexico could be likened to the x-axis on a two-dimensional Cartesian plane.

Similarly, the north-south boundary line between Colorado/New Mexico and Utah/Arizona could be likened to the y-axis on that same Cartesian plane.

The point at the Four Corners would thus be the origin on this two-dimensional Cartesian plane.

My question is: Do Colorado and Arizona touch each other? / Do Utah and New Mexico touch each other?

Think about it: If Colorado and Arizona were to be touching each other at the origin point, how could Utah and New Mexico touch each other across the obstacle which would be formed by the Colorado-Arizona connection?

Sorry bayoubill! Looks like no one can answer your question.

Get a square piece of paper 8x8 using whatever measurement system you want.
Mark the midpoint on each side.
Connect top point to bottom point and the side points to each other.
Label each square like this: top left is A, top right is B, lower left is C, lower left is D.

Do you see the point where all the squares touch? Side by side and diagonally?

Rename the squares to match the names of the states. There's your answer.

I'm not sure how - or whether - the definition of a point differs in Cartesian and Euclidean systems.

However, in the Euclidean system (as I remember it) a point has position but no magnitude. Therefore the point known as Four Corners has an area of zero, and all four states that meet there will touch there.

To answer the apparent paradox: both pairs can touch each other without interfering because CO/NM and AZ/UT would form externally connected topological spaces.

To me, this wasn't a complicated question that deliberately asked in a way to make it appear complicated.

Looking at an X answers the question. I don't even understand what the "apparent paradox" is supposed to be.

The paradox is that if upper right touches lower left then there is a physical continuation between those two quadrants. If those two quadrants form a continuity, doesn't that separate the upper left from the lower right and so they cannot touch.

The problem is that it is presented as a dichotomical either/or, when in fact it is a trichotomy. While upper right is touching lower left, it is simultaneously touching the other two quadrants as well. So 'A' touches 'C' doesn't end the logical proposal. That unfairly mis-describes the situation of 'B' and 'D'. The real way to describe the situations is that 'A' touches 'B', 'C' and 'D'. 'B' touches 'A', 'C' and 'D'. 'C' touches 'A', 'B' and 'D'. 'D' touches 'A', 'B' and 'C'. Now, please excuse me while I ream out my auditory canals real deep with an electric mixer.

e/adition: To say that 'A' only touches 'C' as is posited, is to define a situation other than a four corner one.