FUZZY ORDERED SETS AND DUALITY FOR FINITE FUZZY
... belongs to E and R(y, x) > 0 (y is a lower bound of x) then y belongs to E (an increasing set is defined in a similar way) [16]. A fuzzy ordered space is a triplet (X, τ, R) , where X is a non empty set, τ is a topology on X and R is a fuzzy order on X. A fuzzy lattice is a fuzzy order (A, R), where ...
... belongs to E and R(y, x) > 0 (y is a lower bound of x) then y belongs to E (an increasing set is defined in a similar way) [16]. A fuzzy ordered space is a triplet (X, τ, R) , where X is a non empty set, τ is a topology on X and R is a fuzzy order on X. A fuzzy lattice is a fuzzy order (A, R), where ...
Fuzzy Regular Generalized Super Closed Set
... Proof: If f(A)≤G where GFRO(Y).Then A≤f-1(G) and hence cl(A) ≤ f-1(G)because A is a fuzzy g-super closed in X. Since f is fuzzy super closed, f(cl(A)) is a fuzzy super closed set in Y. It follows that cl(f(A))≤ cl(f(cl(A)))= f(cl(A))≤G. Thus cl(f(A)≤G and f(A) is a fuzzy rg-super closed set in Y. ...
... Proof: If f(A)≤G where GFRO(Y).Then A≤f-1(G) and hence cl(A) ≤ f-1(G)because A is a fuzzy g-super closed in X. Since f is fuzzy super closed, f(cl(A)) is a fuzzy super closed set in Y. It follows that cl(f(A))≤ cl(f(cl(A)))= f(cl(A))≤G. Thus cl(f(A)≤G and f(A) is a fuzzy rg-super closed set in Y. ...
Algebraic Topology
... To avoid overusing the word ‘continuous’ we adopt the convention that maps between spaces are always assumed to be continuous unless otherwise stated. ...
... To avoid overusing the word ‘continuous’ we adopt the convention that maps between spaces are always assumed to be continuous unless otherwise stated. ...
Limit Spaces with Approximations
... Suppose that R Ď X ˆ X N . If limpRq Ď X ˆ X N is defined inductively by the following clauses: (i) R Ď limpRq and tpx, xq | x P X u Ď limpRq, (ii) limpRqpx, xn q Ñ limpRqpx, xαpnq q, for each α P S, (iii) @αPS DβPS plimpRqpx, xαpβpnqq qq Ñ limpRqpx, xn q, then limpRq is the smallest limit relation ...
... Suppose that R Ď X ˆ X N . If limpRq Ď X ˆ X N is defined inductively by the following clauses: (i) R Ď limpRq and tpx, xq | x P X u Ď limpRq, (ii) limpRqpx, xn q Ñ limpRqpx, xαpnq q, for each α P S, (iii) @αPS DβPS plimpRqpx, xαpβpnqq qq Ñ limpRqpx, xn q, then limpRq is the smallest limit relation ...
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. Intuitively, a 3-manifold can be thought of as a possible shape of the universe. Just like a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.