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In topology and related branches of mathematics, a T 1 space is a topological space in which, for every pair of distinct points, each has a neighborhood not containing the other point. [1] An R 0 space is one in which this holds for every pair of topologically distinguishable points. The properties T 1 and R 0 are examples of separation axioms.
The T 0 axiom is special in that it can not only be added to a property (so that completely regular plus T 0 is Tychonoff) but also be subtracted from a property (so that Hausdorff minus T 0 is R 1), in a fairly precise sense; see Kolmogorov quotient for more information. When applied to the separation axioms, this leads to the relationships in ...
Many properties of t-conorms can be obtained by dualizing the properties of t-norms, for example: For any t-conorm ⊥, the number 1 is an annihilating element: ⊥(a, 1) = 1, for any a in [0, 1]. Dually to t-norms, all t-conorms are bounded by the maximum and the drastic t-conorm:
Hausdorff spaces are T 1, meaning that each singleton is a closed set. Similarly, preregular spaces are R 0. Every Hausdorff space is a Sober space although the converse is in general not true. Another property of Hausdorff spaces is that each compact set is a closed set.
Property (T) is preserved under quotients: if G has property (T) and H is a quotient group of G then H has property (T). Equivalently, if a homomorphic image of a group G does not have property (T) then G itself does not have property (T). If G has property (T) then G/[G, G] is compact. Any countable discrete group with property (T) is finitely ...
Following the same procedure with S and T reversed, one finds exactly the same formula, proving that tr(S ∘ T) equals tr(T ∘ S). The above proof can be regarded as being based upon tensor products, given that the fundamental identity of End(V) with V ⊗ V ∗ is equivalent to the expressibility of any linear map as the sum of rank-one ...
A material property is an intensive property of a material, i.e., a physical property or chemical property that does not depend on the amount of the material. These quantitative properties may be used as a metric by which the benefits of one material versus another can be compared, thereby aiding in materials selection.
For a property R that changes when the temperature changes by dT, the temperature coefficient α is defined by the following equation: d R R = α d T {\displaystyle {\frac {dR}{R}}=\alpha \,dT} Here α has the dimension of an inverse temperature and can be expressed e.g. in 1/K or K −1 .