Examples and Counterexamples
The majority of 'everyday' spaces in mathematics are first-countable. In particular, every metric space is first-countable. To see this, note that the set of open balls centered at x with radius 1/n for integers n > 0 form a countable local base at x.
An example of a space which is not first-countable is the cofinite topology on an uncountable set (such as the real line).
Another counterexample is the ordinal space ω1+1 = where ω1 is the first uncountable ordinal number. The element ω1 is a limit point of the subset does not have a countable local base. The subspace ω1 = [0,ω1) is first-countable however, since ω1 is the only such point.
The quotient space where the natural numbers on the real line are identified as a single point is not first countable. However, this space has the property that for any subset A and every element x in the closure of A, there is a sequence in A converging to x. A space with this sequence property is sometimes called a Fréchet-Urysohn space.
Read more about this topic: First-countable Space
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)