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Last updated Nov 1, 2022
...Then D⊂XD \subset XD⊂X is Dense if the of DDD is XXX. Properties DDD is XXX is the only closed se containing DDD The of the of DDD is empt ∀x∈X\forall x \in X∀x∈X, either......
11/7/2022
Statement Suppose VVV is a of R\mathbb{R}R and f:V→Rf: V \to \mathbb{R}f:V→R. Then : fff is a The of fff, Epi(f)⊂V×R\text{Epi}(f) \subset V \times \mathbb{R}Epi(f)⊂V×R (endowed with the ), is a . (Check this - ) :......
... for XXX. Suppose V⊂XV \subset XV⊂X is . Then for each p∈Vp \in Vp∈V, Up∩V∈BU_{p} \cap V \in \mathcal{B}Up∩V∈B. Then we have ⋃p∈VUp∩V=(⋃p∈VUp)∩V=V.\bigcup\limits{p \in V} U{p} \cap V = (\bigcup\limits{p \in V}U{p}) \cap V = V.⋃p∈VUp∩V=(⋃p∈VUp)∩V=V. Therefore, B\mathcal{B}B is......
...> n-1] \cap ([\nu > n])^{C} \in \mathcal{B}_{n},$$ where $[\nu > 0] = \Omega$. $\checkmark$ $\blacksquare$ Remarks Condition (3) is the same as saying [ν≥n+1]∈Bn[\nu \geq n+1] \in \mathcal{B}_{n}[ν≥n+1]∈Bn ∀n∈N\forall n \in \mathbb{N}∀n∈N. Encounters Sect 10.7 pg 365 ......
...exists because MMM is . Let p^:=φp(p)\hat{p} := \varphi{p}(p)p^:=φp(p). Since U^:=φp(Up)\hat{U} := \varphi{p}(U{p})U^:=φp(Up) is in Rn\mathbb{R}^{n}Rn, by definition of the , there exists some radius ϵ>0\epsilon > 0ϵ>0 so Bϵ(p^)⊂U^B{\epsilon}(\hat{p}) \subset \hat{U}Bϵ(p^)⊂U^. Since φp\varphi{p}φp is a (by definition......
...be an L\mathcal{L}L-. Let M\mathcal{M}M be an L\mathcal{L}L-structur Then : M⊨σϕ\mathcal{M} \models_{\sigma} \phiM⊨σϕ for some σ\sigmaσ M⊨σϕ\mathcal{M} \models_{\sigma} \phiM⊨σϕ for all s σ\sigmaσ Proof - (2) implies (1) easily. (1) implies (2) by the . See Corollary......