Any isolated point of ? ( " T " ) is both open and closed in the subspace topology and therefore has an associated spectral projection.
22.
Other examples of disconnected spaces ( that is, spaces which are not connected ) include the plane with an subspace topology induced by two-dimensional Euclidean space.
23.
One can also show that, for each " i ", the subspace topology " X i " inherits from ? coincides with its original topology.
24.
Thus every subset " K " of ? ( " T " ) that is both open and closed in the subspace topology has an associated spectral projection given by
25.
If the disk is viewed as its own topological space ( with the subspace topology of "'R "'2 ), then the boundary of the disk is empty.
26.
The adele ring does "'not "'have the subspace topology, because otherwise the adele ring would not be a locally compact group ( see the theorem below ).
27.
It transpires that this scenario is possible if and only if " K " is both open and closed in the subspace topology on ? ( " T " ).
28.
We give a topology by giving it the subspace topology as a subset of ( where is the space of paths in which as a function space has the compact-open topology ).
29.
This topology is defined by giving the inertia subgroup its subspace topology and imposing that it be an open subgroup of the Weil group . ( The resulting topology is " locally profinite " .)
30.
A space is locally connected if and only if for every open set " U ", the connected components of " U " ( in the subspace topology ) are open.
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