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...

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...

Linear subspace – In mathematics, vector subspace Pointless topology Quasitopological space – Function in topology Relatively compact subspace – Subset...

Hermiticity, 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...

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

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...

{\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...

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...

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...

}(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 ...

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

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...

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

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...

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...

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...

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

between normed spaces which is not bounded below on any infinite-dimensional subspace. Let X and Y be normed linear spaces, and denote by B(X,Y) the space of...

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...

semi-Montel space or perfect if every bounded subset is relatively compact. A subset of a TVS is compact if and only if it is complete and totally bounded....

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....

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...

properties are equivalent: A {\displaystyle A} is equicontinuous; relatively weakly compact; strongly bounded; weakly bounded. The 0-neighborhood bases in...

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...

{\displaystyle B} is a vector subspace of X {\displaystyle X} then so too is B ∘ {\displaystyle B^{\circ }} a vector subspace of Y . {\displaystyle Y.} If...