Changes

Let <math>P_i</math> be an infinite sequence of ever-decreasing open intervals:

Let <math>P_i</math> be an infinite sequence of ever-decreasing open intervals:

::<math>P_i = \{ x\ |\ -1 - 1/i < 1+1/i \}\,</math>

::<math>P_i = \{ x\ |\ -1-1/i < x < 1+1/i \}\,</math>

for integer <math>i \ge 1</math>

for integer <math>i \ge 1</math>

The intersection of all of the <math>P_i</math>'s is the closed interval

The intersection of all of the <math>P_i</math>'s is the closed interval

Proof:  We need a neighborhood of each point in the space.  The neighborhood centered on that point, with radius 1, will do the trick.

Proof:  We need a neighborhood of each point in the space.  The neighborhood centered on that point, with radius 1, will do the trick.

This means that the real line is open.  It is '''not''' an open interval, because that interval would have to be "<math>(-\infty, \infty)</math>, and '''infinity is not a number'''.  The real line is an open '''set''', because it is:

This means that the real line is open.  It is '''not''' an open interval, because that interval would have to be "<math>(-\infty, \infty)</math>", and '''infinity is not a number'''.  The real line is an open '''set''', because it is:

::<math>\mathbb{R}^1 = \bigcup_{n}\ (n-1, n+1)</math>

::<math>\mathbb{R}^1 = \bigcup_{n}\ (n-1, n+1)</math>

over all integers <math>n</math>.  (Infinite unions are allowed, even though infinity is not a number.)

over all integers <math>n</math>.  (Infinite unions are allowed, even though infinity is not a number.)