a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact. Every...

space Paracompact space Quasi-compact morphism Precompact set - also called totally bounded Relatively compact subspace Totally bounded Let X = {a, b}...

These are compact only if they are finite. All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology...

Precompact set may refer to: Relatively compact subspace, a subset whose closure is compact Totally bounded set, a subset that can be covered by finitely...

complete. Compact space Locally compact space Measure of non-compactness Orthocompact space Paracompact space Relatively compact subspace Sutherland...

wikidata descriptions as a fallback Relatively compact subspace – subset of a topological space whose closure is compactPages displaying wikidata descriptions...

bounded subsets of X {\displaystyle X} to relatively compact subsets of Y {\displaystyle Y} (subsets with compact closure in Y {\displaystyle Y} ). Such...

Baire space Banach–Mazur game Meagre set Comeagre set Compact space Relatively compact subspace Heine–Borel theorem Tychonoff's theorem Finite intersection...

In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert...

finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it...

G were compact then there is a unique decomposition of H into a countable direct sum of finite-dimensional, irreducible, invariant subspaces (this is...

x^{-1}\right).} However, if the unit group is endowed with the subspace topology as a subspace of R , {\displaystyle R,} it may not be a topological group...

uniformly on each compact subset of X {\displaystyle X} . Let C c ( X , Y ) {\displaystyle {\mathcal {C}}_{c}(X,Y)} be the subspace of F ( X , Y ) {\displaystyle...

subspace of L2(D); in fact, it is a closed subspace, and so a Hilbert space in its own right. This is a consequence of the estimate, valid on compact...

with the subspace topology induced on it by f {\displaystyle f} 's codomain Y . {\displaystyle Y.} Every strongly open map is a relatively open map....

\ K:={\overline {\operatorname {co} }}S\ } of this compact subset is compact. The vector subspace   X := span ⁡ S = span ⁡ {   e 1 , e 2 , …   }   {\displaystyle...

  K n − 1   {\displaystyle \ {\mathcal {K}}^{n-1}\ } is an invariant subspace of A. To see that, consider any   k ∈ K n − 1 {\displaystyle \ k\in {\mathcal...

}(U)} is endowed with the subspace topology induced on it by C i ( U ) . {\displaystyle C^{i}(U).} If the family of compact sets K = { U ¯ 1 , U ¯ 2 ...

number lemma, which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover. Unlike in the...

proper if f − 1 ( C ) {\displaystyle f^{-1}(C)} is a compact set in X for any compact subspace C of Y. Proximity space A proximity space (X, d) is a...

{\displaystyle K} of C ( X ) {\displaystyle {\mathcal {C}}(X)} is relatively compact if and only if it is bounded in the norm of C ( X ) , {\displaystyle...

topological space X {\displaystyle X} is called relatively compact if its closure is a compact subspace of X . {\displaystyle X.} For any topological space...

is compact. The restriction of T {\displaystyle T} to C c ∞ ( U ) , {\displaystyle C_{c}^{\infty }(U),} when that space is equipped with the subspace topology...

if X if infinite it is not weakly countably compact. Locally compact but not locally relatively compact. If x ∈ X {\displaystyle x\in X} , then the set...

where the relatively weak Zariski topology causes many technical complications. Non-compact topological groups — The class of non-compact groups is too...

relation to a locally compact abelian group G becomes that of a function F in L∞(G), such that its translates by G form a relatively compact set. Equivalently...

defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra. For compactness, we'll use a single capital letter...

of a complete Hausdorff space is compact if (and only if) it is closed and totally bounded. Importantly, the subspace topology that X ′ {\displaystyle...

is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of...

and only if for every open set U, the connected components of U (in the subspace topology) are open. It follows, for instance, that a continuous function...