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I'm not really interested in the actual number of times, but rather an explanation of a closed Universe. How do we still measure the background radiation of the Big Bang? Those electromagnetic waves should have all passed this point in space a long time ago. Unless the Universe is closed and those Electromagnetic waves pass over the same region of space over and over again (like a bug running around the hemisphere of an expanding ball). If I recall correctly, at this point in time, the (circumference) of the Universe is increasing faster than the speed of light so it would be impossible for any current photons to ever complete another lap, but was this always so? Is the background radiation also continuously lowering in frequency as it red-shifts through ever expanding space?

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Yes, the cosmic microwave background frequency has been dropping since its birth 380,000 years after the Big Bang, and it will continue to do so forever.

And yes, it was always impossible for the photons to complete a "lap". In fact, it was even more impossible a short time after the birth of the CMB about 380,000 years after the Big Bang, because the expansion was making the task of completing a lap increasingly hard.

This fact may be seen on the Penrose "causal" diagram of our Universe: its height is actually finite and smaller than the width. This fact is the source of the so-called "horizon problem": because the photons don't have enough time for a round trip, it's hard to understand why all places in the Universe agreed upon a common temperature. Cosmic inflation solves this problem.

In the text above, I assumed that there are no "spooky periodicities" in the visible Universe. It had been proposed that the shape of the Universe was an orbifold defined by a dodecahedron. However, such an assumption implies predictions – we should be able to see certain objects many times in the skies – and no such periodic multiple images have been found which imposes strong constraints on such models.

Finally, I need to disagree with a sentence of yours:

Those electromagnetic waves should have all passed this point in space a long time ago.

That isn't so. The CMB was created at every point of the Universe about 380,000 years after the Big Bang. Right now, we are observing photons that were created exactly at the right place so that they needed those "13.7 billion years minus 380,000 years" to get from the place of birth to the point where our gadgets are located now. As the Universe is getting older, we are observing photons that were created at increasingly distant places of the Universe, but whatever the age of the Universe is, we always find some places with the right distance so that the time needed for the photons to get here exactly agrees with (any) age of the Universe.

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Correction: CMB started 380,000 after the Big Bang, not 380,000 years ago. –  user6975 Jan 4 '12 at 4:54
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