Definition
An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. It is often useful to embed topological spaces in compact spaces, because of the special properties compact spaces have.
Embeddings into compact Hausdorff spaces may be of particular interest. Since every compact Hausdorff space is a Tychonoff space, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.
The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.
Read more about this topic: Compactification (mathematics)
Famous quotes containing the word definition:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)