One could detect that he lives in a closed (i.e. finite) universe by looking for images of himself! That is, we could look for, say, an image of the Virgo cluster appearing far off in the universe. Or, we could look for two images of a given quasar in opposite directions of the sky. However, given the scale, these objects simply change too quickly for us to recognize them at the necesarily dramatically different stages of their evolution. We must take a less direct approach.
Consider the cosmic microwave background (CMB). To us, it appears as a sphere of radiation (called the "surface of last scattering") with us at the center. In a closed universe, this sphere could intersect itself and the intersection would result in a circle (see Figure 11). It might be possible, then, to see the same circles of intersection in two different areas of the sky. Since the CMB has small temperature fluctuations (as revealed by the Cosmic Background Explorer [COBE], see Figure 12), these circles could be detected by looking for similar patterns in the fluctuations. With these circles in hand, a precise model of the universe could be empirically constructed which reflects not only the local geometry of the universe, but also its global topology.
Unfortunately, the resolution of the COBE map is insufficient for detection of the circles (unless the fundamental domain is very small). However, with the expected launch of the Microwave Anistropy Probe (MAP) in late 2000 and the European Space Agency's Planck Explorer in the middle of the next decade, high resolution maps of the CMB should become available. This may allow us to answer the fundamental question "What is the global topology of our universe?" sometime within the next 10 years!
