On Upper and Lower Weakly c-e-Continuous Multifunctions 1
... Proof: Similar to the proof of Theorem 3.3. Definition 3.18 Let A be a subset of a topological space X. The e-frontier of A, denoted by ef r(A), is defined by ef r(A) = e(A)∩e(X\A) = e-(A)\e-(A). Theorem 3.19 Let F : X → Y be a multifunction. The set of all points x of X such that F is not upper wea ...
... Proof: Similar to the proof of Theorem 3.3. Definition 3.18 Let A be a subset of a topological space X. The e-frontier of A, denoted by ef r(A), is defined by ef r(A) = e(A)∩e(X\A) = e-(A)\e-(A). Theorem 3.19 Let F : X → Y be a multifunction. The set of all points x of X such that F is not upper wea ...
- Iranian Journal of Fuzzy Systems
... (A, τ A ) is called fuzzy subspace of (X, τ ). Definition 1.5. Let (X, τ ) be a fuzzy topological space. If all constant fuzzy sets in X are τ −fuzzy open, then (X, τ ) is called fully stratified space. Definition 1.6. Let X and Y be two sets, f : X → Y be a function and A be a fuzzy set in X, B be ...
... (A, τ A ) is called fuzzy subspace of (X, τ ). Definition 1.5. Let (X, τ ) be a fuzzy topological space. If all constant fuzzy sets in X are τ −fuzzy open, then (X, τ ) is called fully stratified space. Definition 1.6. Let X and Y be two sets, f : X → Y be a function and A be a fuzzy set in X, B be ...
Full paper - New Zealand Journal of Mathematics
... (3) If (X, τ ) is a Baire space, then (X, τ ) is weakly Lindelof iff (X, τ ) is M (τ )Lindelof. Proof. (1) Necessity. Assume (X, τ ) is weakly Lindelof and let U be an open cover of X. Then by assumption there exists a countable subcollection V of U such that X = Cl(∪V). X − ∪V is then a closed set ...
... (3) If (X, τ ) is a Baire space, then (X, τ ) is weakly Lindelof iff (X, τ ) is M (τ )Lindelof. Proof. (1) Necessity. Assume (X, τ ) is weakly Lindelof and let U be an open cover of X. Then by assumption there exists a countable subcollection V of U such that X = Cl(∪V). X − ∪V is then a closed set ...
General Topology Pete L. Clark
... Proof. We will use Darboux’s Integrability Criterion: we must show that for all > 0, there exists a partition P of [a, b] such that U (f, P) − L(f, P) < . It is convenient to prove instead the following equivalent statement: for every > 0, there exists a partion P of [a, b] such that U (f, P) − ...
... Proof. We will use Darboux’s Integrability Criterion: we must show that for all > 0, there exists a partition P of [a, b] such that U (f, P) − L(f, P) < . It is convenient to prove instead the following equivalent statement: for every > 0, there exists a partion P of [a, b] such that U (f, P) − ...
The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure B
... 1.b The main theorem The following theorem summarizes the principle results of this article: Theorem 1.8 Let C S 1 (B 3 = f0g) be a nite energy, pseudo-holomorphic subvariety. Then C has nite area and each point in its closure in S 1 B 3 has well de ned tangent cones up to the rotation ! ...
... 1.b The main theorem The following theorem summarizes the principle results of this article: Theorem 1.8 Let C S 1 (B 3 = f0g) be a nite energy, pseudo-holomorphic subvariety. Then C has nite area and each point in its closure in S 1 B 3 has well de ned tangent cones up to the rotation ! ...
