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See Topological space. Totally bounded A metric space M is totally bounded if, for every r > 0, there exist a finite cover of M by open balls of radius r. A metric space is compact if and only if it is complete and totally bounded. Totally disconnected A space is totally disconnected if it has no connected subset with more than one point ...
[0, 1] 2 is a totally bounded space because for every ε > 0, the unit square can be covered by finitely many open discs of radius ε. A metric space (,) is totally bounded if and only if for every real number >, there exists a finite collection of open balls of radius whose centers lie in M and whose union contains M.
Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space. The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpiński space is compact. No discrete space with an infinite number of points is ...
If Y is a Banach space, an equivalent definition is that the embedding operator (the identity) i : X → Y is a compact operator. When applied to functional analysis, this version of compact embedding is usually used with Banach spaces of functions. Several of the Sobolev embedding theorems are compact embedding theorems.
In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space R n or the complex coordinate space C n. A connected open subset of coordinate space is frequently used for the domain of a function. [a]
A subset S of a metric space (M, d) is bounded if there exists r > 0 such that for all s and t in S, we have d(s, t) < r. The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent.
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A definition "from scratch", as in Euclid, is now not often used, since it does not reveal the relation of this space to other spaces. Also, a three-dimensional projective space is now defined as the space of all one-dimensional subspaces (that is, straight lines through the origin) of a four-dimensional vector space. This shift in foundations ...