Author of more than 225 research publications

Has successfully supervised 36 graduate students in their work towards a Ph.D. degree.  His former Ph.D. students nowadays work as professors in universities located in the United States, Canada, Japan, Mexico, Russia and Bulgaria.

A.V.Arhangel'skii's scientific research feature a wide-range approach to problems embracing various aspects of General Topology --- thus, reciprocal classification of spaces and maps, theory of cardinal invariants of topological spaces, topological algebra, spaces of continuous functions. A.V.Arhangel'skii's works involve new interesting concepts and a great deal of new important problems. By this reason many researchers follow his studies. It is not an exaggeration to claim that the research in various fields of General Topology carried out all over the World, was considerably stimulated by A.V.Arhangel'skii's works.

The most famous A.V.Arhangel'skii's result is the solution of the old problem of P.S.Alexandroff (dated to 1922) on the cardinality of a compact Hausdorff first-countable space, in 1969. A.V.Arhangel'skii introduced such fundamental concepts as a network of a topological space, tightness, free sequence; proved the coincidence of weight and network weight in compacta and, on its basis, the addition theorem for weight of compacta; proved that the supremum of lengths of free sequences in a compactum is equal to its tightness; performed a systematic investigation of various classes of maps --- open, pseudo-open, quotient, continuous bijections --- in objective to characterize topological properties of spaces; defined and studied the class of $p$-spaces (or `feathered' spaces) which is a common generalization of metrizable and Cech-complete spaces; gave the characterization of paracompact $p$-spaces as perfect preimages of metrizable spaces; introduced the class of symmetrizable spaces; proved a metrization theorem based on the concept of a regular base which is also due to him. A.V.Arhangel'skii has made a dominant contribution to the foundation of $C\_p\_$-theory whose subject is the study of the space $C\_p\_(X)$ of continuous functions on the space $X$, with the topology of pointwise convergence, particularly its properties in relation with properties of $X$.

Some new areas developed by A.V.Arhangel'skii and attracting still more investigators, are the theory of relative topological properties and the theory of cleavable and weakly normal spaces. Arhangel'skii's scientific school is now actively studying problems of topological homogeneity of spaces, in particular, some generalizations of topological groups (rectifiable spaces, spaces with a Maltsev operation --- an `anti-mixer').

The following recent scientific results of A.V.Arhangel'skii should be mentioned: any compact space cleavable over the real line embeds into it; a compact topological group of a non Ulam-measurable cardinality has no stronger countably compact topology; a topological group having a dense subspace with countable (absolute) tightness is metrizable; if $C\_p\_(X)$ has countable spread then it admits a continuous bijection onto a Tychonoff space whose finite powers are all hereditarily separable.

At present, A.V. Arhangel'skii works on a broad range of problems in topology and its applications.  They include the theory of cardinal invariants of topological groups, the study of homogeneous compacta, the diagonal properties of spaces in connection with the study of metrizability type properties, the theory of compactification in interaction with topological groups theory, dulity theorems about remainders in compactification.

Only in the  last 5 years Arhangel'skii published more than 25 papers on his research.