I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
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Actually, the commenter is exactly right. The real line does contain the open interval (0,1). The open interval (0,1) has the exact same cardinality as the real numbers.
An easy map that uniquely maps a real number to a number of the interval (0,1) is the function mapping x to arctan(x)/π + 1/2. The existence of a bijection proves that the sets have the same size, despite one wholly containing the other.
The comment, like the meme, plays on the difference between common intuition and mathematical intuition.