Topologically Complete Spaces
Note that completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval (0, 1), which is not complete. Another example is given by the irrational numbers, which are not complete as a subspace of the real numbers but are homeomorphic to NN (see the sequence example in Examples above).
In topology one considers topologically complete (or completely metrizable) spaces, spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space. Since the conclusion of the Baire category theorem is purely topological, it applies to these spaces as well.
A topological space homeomorphic to a separable complete metric space is called a Polish space.
Read more about this topic: Complete Metric Space
Famous quotes containing the words complete and/or spaces:
“Thus when I come to shape here at this table between my hands the story of my life and set it before you as a complete thing, I have to recall things gone far, gone deep, sunk into this life or that and become part of it; dreams, too, things surrounding me, and the inmates, those old half-articulate ghosts who keep up their hauntings by day and night ... shadows of people one might have been; unborn selves.”
—Virginia Woolf (1882–1941)
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)