Compactification : Stone Cech compactification
It is very useful to embed topological spaces in compact spaces, because of the strong properties compact spaces have. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X.Of particular interest are Hausdorff compactifications, i.e., compactifications in which the compact space is Hausdorff. A topological space has a Hausdorff compactification if and only if it is Tychonoff. Moreover, there is a unique (up to homeomorphism) "most general" compactification, the Stone-Čech compatification of X, denoted by βX. The space βX is characterized by the universal property that any continuous function from X to a compact Hausdorff space K can be extended to a continuous function from βX to K in a unique way.
Any non-compact space X has a one-point compactification obtained by adding an extra point ∞ (often called a point at infinity) and defining the open sets of the new space to be the open sets of X together with the sets of the form G U {∞}, where G is open and X \ G is compact. The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact.
In the String theory context, Compactification refers to "curling" up the extra dimensions ( six in the superstring theory) usually on Calabi-Yau spaces or on Orbifolds[?]
And how strangely his
all, but what would actually be the fate of each individual
would all go to his wife, of course, and how little he cared to
Rose Mall twenty-five thousand or so--so she could always be
him if he got to driving her too hard, and crushing out the
understood, and he had hurt her enough, in all conscience. The
that he didn't care who got it. She could leave it to whomever
happen to it--that was what concerned him.
By his side, Rose, sitting so motionless that he was scarcely
gift of profoundly sympathetic natures she was thinking and
heart-breaking that Martin must be forced to abandon the only
of Sharon for them--she had never been in any doubt as to the
before. And she knew his one thought now must be for their
so little real brightness out of life. She felt, generously, with
suffered through him--from his first broken promises about the
resulted in Billy's death--with even that, she had salvaged so
within her; all the black moments, all the long, gray hours of
again as he had been the day he had proposed marriage to her and
puzzling look that had come into his eyes when she had asked him
about happiness? With a noiseless sob, she remembered that he had
And she saw him again, too, as he had been the night he had so
free of all rancor, unutterably tragic that the only person
for herself. He should never have asked her to marry.
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