General Topology

adherence value B₁ ball with center Banach space bijective Cauchy sequence closed ball closed subset compact space compact subset conditions are equivalent contained continuous mapping Corollary d(xo defined Definition denoted E₁ element example exists a neighborhood exists an open f is continuous f tends F₁ filter base finite number finite subset following conditions fundamental system ƒ a mapping Hilbert space homeomorphism intersection Let f Let us show Let xo linear subspace locally compact space lower semicontinuous normed space numerical function open ball open neighborhood open set open subset pre-Hilbert space radius real numbers real-valued functions resp S₁ satisfied say that ƒ scalar product separated pre-Hilbert space set equipped Suppose condition supxex system of neighborhoods Theorem topological space U₁ U₂ uniform convergence V₁ value of ƒ vector space x₁ Xn+1 y₁