Search results
Results from the WOW.Com Content Network
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.
The Stone–Čech compactification can be constructed explicitly as follows: let C be the set of continuous functions from X to the closed interval [0, 1]. Then each point in X can be identified with an evaluation function on C. Thus X can be identified with a subset of [0, 1] C, the space of all functions from C to [0, 1].
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.
In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in Y has a subsequence convergent in X. Some major theorems characterize relatively compact subsets, in particular in function spaces. An example is the Arzelà–Ascoli theorem.
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).
A space is pseudocompact if every continuous real-valued function on the space is bounded. σ-compact. A space is σ-compact if it is the union of countably many compact subsets. Lindelöf. A space is Lindelöf if every open cover has a countable subcover. Paracompact. A space is paracompact if every open cover has an open locally finite ...