...space-time) is unstable with respect to a (still to be specified) natural topology of space-times [[#References|[a9]]], [[#References|[a10]]]. Here, "future i
5 KB (717 words) - 15:40, 19 February 2021
...TR><TD valign="top">[4]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)</TD></TR></table>
2 KB (272 words) - 13:05, 24 December 2020
...ology compatible with the ordinary topology on all its simplices; the weak topology may serve as an example.
In topology and differential geometry, these complexes occur most frequently as realiza
3 KB (419 words) - 17:16, 7 February 2011
topology $ \sigma ( E ^ \prime , E) $.
topology (the Banach–Alaoglu theorem).
6 KB (887 words) - 08:06, 6 June 2020
...ial role, and often a key one, in constructions in all branches of general topology and in many of its applications. In particular, it is of fundamental import
...role — in particular, in functional analysis (Banach spaces with the weak topology, measures on topological spaces), in the general theory of optimal control,
3 KB (412 words) - 17:06, 7 May 2016
Path-connected spaces play an important role in homotopic topology. If a space $ X $
...TR><TD valign="top">[1]</TD> <TD valign="top"> E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)</TD></TR></table>
3 KB (480 words) - 10:17, 19 January 2022
...etermined by the nature of these concepts. The initial concepts of general topology are the concepts of a [[topological space]] and a [[continuous mapping]], i
...ures defined on the set of points of the space, in some way related to its topology. The topological invariance of the homology groups may serve as an example.
10 KB (1,522 words) - 19:35, 25 March 2023
that converges weakly (cf. [[Weak topology|Weak topology]]) to a [[Probability measure|probability measure]] $ P $(
3 KB (375 words) - 08:14, 6 June 2020
...at it is possible to construct an equivalent metric (i.e., giving the same topology) under which the space $\mathcal{D}$ becomes a separable and complete metri
...D}$ is called the Skorokhod space (cf. also [[Skorokhod topology|Skorokhod topology]]). This space is very important in the theory of random processes (cf. als
8 KB (1,131 words) - 14:43, 27 January 2024
...–Borel structure contains all sets of the Borel structure generated by the topology of $ \widehat{A} $; each point of $ \widehat{A} $ is a Borel set in the Mac
...isomorphic, as a Borel structure, to the Borel structure generated by the topology of some complete separable metric space).
3 KB (478 words) - 04:32, 4 December 2016
...g on which mappings $X \rightarrow Y$ are included in $F$ and what natural topology $F$ is endowed with. The choice of $F$ is related to the presence of additi
...ing way. A family $S$ of subsets of $X$ is fixed, and a [[pre-base]] for a topology $\mathfrak{T}$ on $F$ is formed by sets of the form
8 KB (1,267 words) - 05:59, 22 April 2023
...teria of reflexivity of a Banach space $X$ is weak compactness (cf. [[Weak topology]]) of the unit ball of this space.
2 KB (354 words) - 18:18, 26 September 2017
...$\mathbf R _ { d}$ be the real line $\mathbf{R}$ endowed with the discrete topology and suppose the algebra $A ( G )$ consists of the functions in $C ( G )$ wh
...ce|Hausdorff space]] upon which the real line $\mathbf{R}$ (with the usual topology) acts as a locally compact [[Transformation group|transformation group]]. T
7 KB (1,114 words) - 19:36, 23 December 2023
...onvergence to topologies on the space which are stronger than the original topology (for references to these results, see the historical survey of [[#Reference
...een made to replace the [[Weak topology|weak topology]] of $X$ by a weaker topology, $\sigma ( X , Y )$, generated by a subspace $Y$ of the dual space of $X$ w
5 KB (714 words) - 15:30, 1 July 2020
...al signed Radon measures and the corresponding topology is called ''narrow topology'' by some authors. Several other notions of convergence can be introduced o
4 KB (739 words) - 10:38, 23 November 2013
The density topology $\mathcal{T}_d$ on $\mathbb R$ consists of the family of all subsets $E\sub
...tion $\mathcal{A} = C (\mathcal{T}_d)$ shows the importance of the density topology in real analysis, since the class $\mathcal{A}$ is strongly tied to the the
6 KB (865 words) - 22:40, 17 August 2013
...city of their definition, makes them important standard objects in general topology. However, the topological structure of the Tikhonov cubes is far from trivi
2 KB (392 words) - 15:51, 11 August 2014
Define a topology on $ X $
This topology is called the hull-kernel topology on $ X $.
7 KB (1,058 words) - 08:29, 6 June 2020
...$ is dense in $T$. If the algebraic dual $E^*$ of $E$, is given the [[weak topology]] (so that $E^* \simeq \prod_{\alpha \in A} K$,where $K$ is the base field
1 KB (185 words) - 19:44, 27 February 2021
...s functions on the interval $[ 0,1 ]$, with the [[Uniform topology|uniform topology]], the weak convergence
6 KB (857 words) - 21:45, 15 December 2020