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...

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...

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...

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...

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...

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

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

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

the sense that each point has a T1 neighbourhood (when given the subspace topology), is also T1. Similarly, a space that is locally R0 is also R0. In...

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...

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...

{\displaystyle S=\{x\}\cup \left\{x_{a}:a\in A\right\}} is endowed with the subspace topology induced on it by X , {\displaystyle X,} then lim x ∙ → x {\displaystyle...

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...

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...

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

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

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...

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...

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}...

{\mathcal {C}}} of subspaces of X . {\displaystyle X.} The space X {\displaystyle X} is then compactly generated exactly when its topology is coherent with...

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...

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

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

topological spaces is disconnected product topology, the dual construction subspace topology and its dual quotient topology topological union, a generalization...

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