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That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is known locally – in a neighbourhood of each point of the space – and to extend it to information that holds globally throughout the ...
a discrete subspace of some given topological space (,) refers to a topological subspace of (,) (a subset of together with the subspace topology that (,) induces on it) whose topology is equal to the discrete topology.
In a topological space, subsets without limit point are exactly those that are closed and discrete in the subspace topology. So a space is limit point compact if and only if all its closed discrete subsets are finite. A space is not limit point compact if and only if it has an infinite closed discrete subspace.
It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ∞. The resulting compactification is homeomorphic to a circle in the plane (which, as a closed and bounded subset of the Euclidean plane, is compact). Every sequence that ran off to infinity in the real line will then ...
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed.A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space).
[14] [15] That is, the conclusion of the Baire category theorem holds: the interior of every countable union of nowhere dense subsets is empty. A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is locally closed in Y (that is, X can be written as the set-theoretic difference of two closed subsets of Y). In ...
For T1 spaces, countable compactness and limit point compactness are equivalent. Every sequentially compact space is countably compact. [4] The converse does not hold. For example, the product of continuum-many closed intervals [,] with the product topology is compact and hence countably compact; but it is not sequentially compact.
If {x n} is bounded, then compactness of C implies that there exists a subsequence x nk such that C x nk is norm convergent. So x nk = (I - C)x nk + C x nk is norm convergent, to some x. This gives (I − C)x nk → (I − C)x = y. The same argument goes through if the distances d(x n, Ker(I − C)) is bounded.