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:
“Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
—Henry David Thoreau (1817–1862)
“Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
—Henry David Thoreau (1817–1862)
“Just as children, step by step, must separate from their parents, we will have to separate from them. And we will probably suffer...from some degree of separation anxiety: because separation ends sweet symbiosis. Because separation reduces our power and control. Because separation makes us feel less needed, less important. And because separation exposes our children to danger.”
—Judith Viorst (20th century)
““I tell you the solemn truth that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the axioms of physics.””
—Henry Brooks Adams (1838–1918)
“The doctrine of those who have denied that certainty could be attained at all, has some agreement with my way of proceeding at the first setting out; but they end in being infinitely separated and opposed. For the holders of that doctrine assert simply that nothing can be known; I also assert that not much can be known in nature by the way which is now in use. But then they go on to destroy the authority of the senses and understanding; whereas I proceed to devise helps for the same.”
—Francis Bacon (1560–1626)
“through the spaces of the dark
Midnight shakes the memory
As a madman shakes a dead geranium.”
—T.S. (Thomas Stearns)