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 (...
9 KB (1,446 words) - 16:51, 12 April 2025
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...
15 KB (2,188 words) - 04:51, 21 July 2025
} is the coarsest topology on X {\displaystyle X} that makes those functions continuous. The subspace topology and product topology constructions are...
18 KB (3,386 words) - 11:46, 2 June 2025
singleton subsets). The topology is coherent with the finite subspaces of X {\displaystyle X} . The inclusion maps of the finite subspaces of X {\displaystyle...
12 KB (1,604 words) - 17:27, 20 July 2025
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...
3 KB (366 words) - 03:25, 11 May 2023
Compact space (redirect from Compact subspace)
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...
45 KB (5,704 words) - 04:39, 27 June 2025
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...
23 KB (4,320 words) - 14:39, 26 May 2025
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...
55 KB (7,693 words) - 07:57, 22 February 2025
Dense set (redirect from Dense subspace)
{\displaystyle B} is dense in C {\displaystyle C} (in the respective subspace topology) then A {\displaystyle A} is also dense in C . {\displaystyle C.}...
12 KB (1,911 words) - 17:24, 17 July 2025
interior. In the space of rational numbers with the usual topology (the subspace topology of R {\displaystyle \mathbb {R} } ), the boundary of ( − ∞...
21 KB (3,803 words) - 22:29, 23 May 2025
Connected space (redirect from Connected (topology))
Connected and disconnected subspaces of R² In topology and related branches of mathematics, a connected space is a topological space that cannot be represented...
27 KB (3,874 words) - 20:36, 24 March 2025
T1 space (redirect from T1 topology)
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...
12 KB (1,928 words) - 18:59, 18 June 2025
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...
41 KB (5,740 words) - 19:21, 12 March 2025
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...
33 KB (4,640 words) - 18:24, 27 July 2025
Net (mathematics) (redirect from Net (topology))
{\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...
46 KB (7,342 words) - 08:45, 29 July 2025
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...
21 KB (3,483 words) - 05:09, 28 June 2025
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...
2 KB (201 words) - 10:11, 27 March 2025
Discrete space (redirect from Discrete 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,...
15 KB (2,288 words) - 20:07, 21 January 2025
equipped with the subspace topology inherited from C ∞ ( U ) {\displaystyle C^{\infty }(U)} (a coarser topology than the canonical LF topology), is continuous;...
106 KB (18,955 words) - 20:23, 21 July 2025
Meagre set (redirect from Meagre subspace)
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...
18 KB (2,923 words) - 23:02, 27 July 2025
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...
23 KB (4,068 words) - 06:15, 25 July 2025
Open set (redirect from Open (topology))
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...
27 KB (4,389 words) - 20:38, 20 October 2024
{\displaystyle T} is a subspace of X {\displaystyle X} (meaning that T {\displaystyle T} is endowed with the subspace topology that X {\displaystyle X}...
26 KB (4,287 words) - 09:15, 20 December 2024
Compactly generated space (redirect from K-space (topology))
{\mathcal {C}}} of subspaces of X . {\displaystyle X.} The space X {\displaystyle X} is then compactly generated exactly when its topology is coherent with...
30 KB (4,652 words) - 15:25, 21 April 2025
Distribution (mathematics) (section Topology on the space of distributions and its relation to the weak-* topology)
A linear subspace of D ′ ( U ) {\displaystyle {\mathcal {D}}'(U)} carrying a locally convex topology that is finer than the subspace topology induced on...
128 KB (21,579 words) - 18:41, 21 June 2025
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...
7 KB (937 words) - 01:19, 28 May 2025
natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which...
13 KB (2,206 words) - 15:10, 10 March 2025
Topological space (redirect from Topology (structure))
Linear subspace – In mathematics, vector subspace Pointless topology Quasitopological space – Function in topology Relatively compact subspace – Subset...
27 KB (3,839 words) - 21:04, 18 July 2025
topological spaces is disconnected product topology, the dual construction subspace topology and its dual quotient topology topological union, a generalization...
4 KB (528 words) - 19:05, 3 June 2025
generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor...
42 KB (6,396 words) - 00:29, 17 July 2025