Properties
One of the most important properties of first-countable spaces is that given a subset A, a point x lies in the closure of A if and only if there exists a sequence {xn} in A which converges to x. This has consequences for limits and continuity. In particular, if f is a function on a first-countable space, then f has a limit L at the point x if and only if for every sequence xn → x, where xn ≠ x for all n, we have f(xn) → L. Also, if f is a function on a first-countable space, then f is continuous if and only if whenever xn → x, then f(xn) → f(x).
In first-countable spaces, sequential compactness and countable compactness are equivalent properties. However, there exist examples of sequentially compact, first-countable spaces which aren't compact (these are necessarily non-metric spaces). One such space is the ordinal space [0,ω1). Every first-countable space is compactly generated.
Every subspace of a first-countable space is first-countable. Any countable product of a first-countable space is first-countable, although uncountable products need not be.
Read more about this topic: First-countable Space
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)