Basic Properties of Metrizable Topological Spaces

We continue Mizar formalization of general topology according to the book [11] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that metrizable topological space. We also define Lindel¨of spaces and state the above theorem in this special case. We also introduce the concept of separation among two subsets (see [12]).Institute of Computer Science, University of Białystok, PolandGrzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Józef Białas and Yatsuka Nakamura. The theorem of Weierstrass. Formalized Mathematics, 5(3):353-359, 1996.Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Ryszard Engelking. General Topology, volume 60 of Monografie Matematyczne. PWN-Polish Scientific Publishers, Warsaw, 1977.Ryszard Engelking. Teoria wymiaru. PWN, 1981.Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55-59, 1999.Adam Grabowski. On the Borel families of subsets of topological spaces. Formalized Mathematics, 13(4):453-461, 2005.Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13(1):139-146, 2005.Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Zbigniew Karno. Maximal discrete subspaces of almost discrete topological spaces. Formalized Mathematics, 4(1):125-135, 1993.Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233-236, 1996.Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

Space-Time Isogeometric Analysis of Parabolic Evolution Equations

We present and analyze a new stable space-time Isogeometric Analysis (IgA) method for the numerical solution of parabolic evolution equations in fixed and moving spatial computational domains. The discrete bilinear form is elliptic on the IgA space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the IgA spaces yields an a priori discretization error estimate with respect to the discrete norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IgA spaces

Computing Invariants of Simplicial Manifolds

This is a survey of known algorithms in algebraic topology with a focus on finite simplicial complexes and, in particular, simplicial manifolds. Wherever possible an elementary approach is chosen. This way the text may also serve as a condensed but very basic introduction to the algebraic topology of simplicial manifolds. This text will appear as a chapter in the forthcoming book "Triangulated Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure