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I noticed when looking at some deep space pictures, like the Sombrero Galaxy (M104) or the Hubble Deep Field (HDF), I see galaxies of various colors. Does that mean, for instance, than any inhabitant of a red-colored galaxy will look up and see a sky full of red stars?

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If the inhabitants have human-like vision, and the density of stars near them is similar to the density of those near the Earth, then probably not, because human eyes do not perceive colors of faint objects very well. See the answers to this question.

Yes, there is a lot of genuine variation in the color of galaxies, but keep in mind that colors in different pictures of astronomical objects are not always directly comparable; see the same question. The images from SDSS found in parts of google sky and on the SDSS CAS are good places for finding sets of images whose colors were generated uniformly.

If our perception of color were better at faint light levels, it would be clear that the stars in our own sky have a variety of colors. In a red galaxy, the ratio of red to blue stars would be higher than in ours. If the inhabitants are in a much denser part of their galaxy (such that they would have bright enough stars close by to see the colors more clearly), or if they have better color perception than ours, maybe they could see a sky with many more red stars.

The colors of populations of stars are strongly related to the ages of the stars. Newly formed populations of stars include more very bright blue stars, while older populations have more bright red ones. (See stellar evolution for why.) So, galaxies that are more blue have more recent star formation, while those that are more red have less. Compare pictures of globular clusters, which have old populations of stars, to those of open clusters, which have younger populations. Note that, at least in globular clusters, even the old populations still have some relatively blue stars (see the blue horizontal branch and blue stragglers).

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I suspect there would be a few bright enough. A lot of the light in for instance globular clusters comes from giant branch stars, with typically a few thousand solar luminosities apiece. Old (spehrical) galaxies, generally haven't had as long a time since starformation stopped as globular clusters, so the mass cutoff limit should be higher. –  Omega Centauri Aug 15 '11 at 21:52

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