Sentence examples for be a closed connected from inspiring English sources

Let ([a,b]) be a compact interval, and let (Yin {mathcal{A}}_{n}) be a closed, connected and locally connected subset of ({mathbf{R}}^{n}).

Let M be a closed, connected, and C ∞ Riemannian manifold endowed with a volume form μ, and let μ denote the Lebesgue measure associated to it.

Let (M,g) be a closed connected Riemannian manifold, L TM→R be a Tonelli Lagrangian.

Let M be a closed, connected and smooth n ( ≥ 1 ) -dimensional Riemannian manifold endowed with a volume form, which has a measure μ, called the Lebesgue measure.

Let Gamma=overline{bigl{ (lambda,V inmathbb{R}times E: V=lambda SV+FV, V neq0bigr} }, and let (mathcal{C}) be a closed connected component of Γ that contains ((lambda,0)).

In the present section we are going to show that M is a closed, connected, real-analytic submanifold of (mathrm {H}(W^s)) which locally agrees with (mathrm {Fix} (psi )).

Let (mathcal{O} subsetmathbb{R}^{n}) be a bounded closed connected domain.

We consider the following parameterized symplectic mapping: Phi(cdot;xi): (x,u,y, v inmathbb{T}^{n}timesmathcal{W} times mathcal{O} timesmathcal{W} to(hat{x},hat{u},hat {y}, hat{v})in mathbb{T}^{n}times mathbb{R}^{m}timesmathbb{R}^{n} timesmathbb{R}^{m}, where (xiinPisubsetmathcal{O}) is a parameter and (mathcal {O} subsetmathbb{R}^{n}) is a bounded closed connected domain.

They are given by the following lemma, which extends Lemma 2.1 of [47] (see [43, 44, 49]): Let (Hsubset mathrm{U}_N) be a connected closed subgroup, with (Nge 2).

Further, in order to investigate lower group dimensions, we need the following result, which can be regarded as a unitary analogue of Lemma 3.5, as well as parts (I), (II) of the lemma stated on p. 347 in [50] (see [41], Lemma 1.4], [46]): Let (Hsubset mathrm{U}_N) be a connected closed subgroup, with (Nge 2).

Since the closure of any connected subset is also connected, each component of (Y) is a closed subset.