induced from that of 𝜏 called the subspace topology (or the relative topology, or the induced topology, or the trace topology). Given a topological space (...

subspace topology on Z, so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology...

} is the coarsest topology on X {\displaystyle X} that makes those functions continuous. The subspace topology and product topology constructions are...

topological space X is said to be compact if it is compact as a subspace (in the subspace topology). That is, K is compact if for every arbitrary collection...

singleton subsets). The topology is coherent with the finite subspaces of X {\displaystyle X} . The inclusion maps of the finite subspaces of X {\displaystyle...

order topology, i.e. the order topology of the totally ordered Y, where this order is inherited from X, is coarser than the subspace topology of the...

submaximal space is irresolvable. Subspace If T is a topology on a space X, and if A is a subset of X, then the subspace topology on A induced by T consists...

many topologies are most easily defined in terms of a base that generates them. Every subset of a topological space can be given the subspace topology in...

{\displaystyle B} is dense in C {\displaystyle C} (in the respective subspace topology) then A {\displaystyle A} is also dense in C . {\displaystyle C.}...

Connected and disconnected subspaces of R² In topology and related branches of mathematics, a connected space is a topological space that cannot be represented...

interior. In the space of rational numbers with the usual topology (the subspace topology of R {\displaystyle \mathbb {R} } ), the boundary of ( − ∞...

In topology, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The...

parent space A subset of a topological space endowed with the subspace topology Linear subspace, in linear algebra, a subset of a vector space that is closed...

equipped with the subspace topology inherited from C ∞ ( U ) {\displaystyle C^{\infty }(U)} (a coarser topology than the canonical LF topology), is continuous;...

natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which...

that the final topology often appears. A topology is coherent with some collection of subspaces if and only if it is the final topology induced by the...

set Y can be given its own topology (called the 'subspace topology') defined by "a set U is open in the subspace topology on Y if and only if U is the...

{\displaystyle T} is a subspace of X {\displaystyle X} (meaning that T {\displaystyle T} is endowed with the subspace topology that X {\displaystyle X}...

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators,...

generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor...

linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when...

different from the subspace topology. For ⁠ 0 < p < ∞ {\displaystyle 0<p<\infty } ⁠, ⁠ ℓ p {\displaystyle \textstyle \ell ^{p}} ⁠ is the subspace of ⁠ K N {\displaystyle...

A linear subspace of D ′ ( U ) {\displaystyle {\mathcal {D}}'(U)} carrying a locally convex topology that is finer than the subspace topology induced on...

in affine N-space. The topology on the adelic algebraic group G ( A ) {\displaystyle G(A)} is taken to be the subspace topology in AN, the Cartesian product...

topology is the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology...

together with the subspace topology that ( Y , τ ) {\displaystyle (Y,\tau )} induces on it) whose topology is equal to the discrete topology. For example,...

algebraic set (irreducible or not) then the Zariski topology on it is defined simply to be the subspace topology induced by its inclusion into some A n . {\displaystyle...

openness condition. Subspace topology: If Y ⊆ X {\displaystyle Y\subseteq X} and X {\displaystyle X} carries the cocountable topology, then Y {\displaystyle...

A} can also be called a meagre subspace of X {\displaystyle X} , meaning a meagre space when given the subspace topology. Importantly, this is not the...

{\displaystyle X} to be endowed with the subspace topology induced on it by, say, the strong dual topology β ( ( X β ′ ) ′ , X β ′ ) {\displaystyle \beta...