Separated Sets - Relation To Separation Axioms and Separated Spaces

Relation To Separation Axioms and Separated Spaces

The separation axioms are various conditions that are sometimes imposed upon topological spaces which can be described in terms of the various types of separated sets. As an example, we will define the T2 axiom, which is the condition imposed on separated spaces. Specifically, a topological space is separated if, given any two distinct points x and y, the singleton sets {x} and {y} are separated by neighbourhoods.

Separated spaces are also called Hausdorff spaces or T2 spaces. Further discussion of separated spaces may be found in the article Hausdorff space. General discussion of the various separation axioms is in the article Separation axiom.

Read more about this topic:  Separated Sets

Famous quotes containing the words relation to, relation, separation, axioms, separated and/or spaces:

    Light is meaningful only in relation to darkness, and truth presupposes error. It is these mingled opposites which people our life, which make it pungent, intoxicating. We only exist in terms of this conflict, in the zone where black and white clash.
    Louis Aragon (1897–1982)

    Every word was once a poem. Every new relation is a new word.
    Ralph Waldo Emerson (1803–1882)

    The scholar was not raised by the sacred thoughts amongst which he dwelt, but used them to selfish ends. He was a profane person, and became a showman, turning his gifts to marketable use, and not to his own sustenance and growth. It was found that the intellect could be independently developed, that is, in separation from the man, as any single organ can be invigorated, and the result was monstrous.
    Ralph Waldo Emerson (1803–1882)

    The axioms of physics translate the laws of ethics. Thus, “the whole is greater than its part;” “reaction is equal to action;” “the smallest weight may be made to lift the greatest, the difference of weight being compensated by time;” and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.
    Ralph Waldo Emerson (1803–1882)

    Science is neither a single tradition, nor the best tradition there is, except for people who have become accustomed to its presence, its benefits and its disadvantages. In a democracy it should be separated from the state just as churches are now separated from the state.
    Paul Feyerabend (1924–1994)

    Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.
    Henry David Thoreau (1817–1862)