Properties of A Pair of Spaces
Given some notion of equivalence (e.g., homeomorphism, diffeomorphism, isometry) between topological spaces, two spaces are locally equivalent if every point of the first space has a neighborhood which is equivalent to a neighborhood of the second space.
For instance, the circle and the line are very different objects. One cannot stretch the circle to look like the line, nor compress the line to fit on the circle without gaps or overlaps. However, a small piece of the circle can be stretched and flattened out to look like a small piece of the line. For this reason, one may say that the circle and the line are locally equivalent.
Similarly, the sphere and the plane are locally equivalent. A small enough observer standing on the surface of a sphere (e.g., a person and the Earth) would find it indistinguishable from a plane.
Read more about this topic: Local Property
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“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)
“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)
“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 Ovarian Theory of Literature, or, rather, its complement, the Testicular Theory. A recent camp follower ... of this explicit theory is ... Norman Mailer, who has attributed his own gift, and the literary gift in general, solely and directly to the possession of a specific pair of organs. One writes with these organs, Mailer has said ... and I have always wondered with what shade of ink he manages to do it.”
—Cynthia Ozick (b. 1928)
“through the spaces of the dark
Midnight shakes the memory
As a madman shakes a dead geranium.”
—T.S. (Thomas Stearns)