Beitr¨ age zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 51 (2010), No. 2, 547-576. Compactness and Boundedness of Tangent Spaces to Metric Spaces O. Dovgoshey F. Abdullayev M. K¨ u¸ c¨ ukaslan Institute of Applied Mathematics and Mechanics of NASU, R. Luxemburg str. 74, Donetsk 83114, Ukraine e-mail: aleksdov@mail.ru Mersin University, Faculty of Literature and Science Department of Mathematics, 33342 Mersin, Turkey e-mail: fabdul@mersin.edu.tr mkucukaslan@mersin.edu.tr Abstract. We describe metric spaces with bounded pretangent spaces and characterize proper metric spaces with proper tangent spaces. We also present the necessary and sufficient conditions under which a tan- gent space is compact and build a compact ultrametric space X such that some pretangent space to X has the density c. MSC 2000: 54E35 Keywords: metric spaces, tangent spaces to metric spaces, compact tangent spaces, bounded tangent spaces, proper tangent spaces 1. Introduction Analysis on metric spaces with no a priori smooth structure is in need of some generalized differentiations. Important examples of such generalizations and even an axiomatics of so-called “pseudo-gradients” can be found in [1], [4], [3], [9], [15], [24], [16] and, respectively, in [2]. Another natural way to obtain suitable dif- ferentiations on metric spaces is to induce some tangents at the points of these space. The Gromov-Hausdorff convergence and the ultra-convergence are, prob- ably, the most widely applied today’s tools for the construction of such tangent spaces (see, for example, [7], [6] and, respectively, [5], [21]). Recently a new ap- proach to the introduction of the tangent spaces at the points of general metric 0138-4821/93 $ 2.50 c  2010 Heldermann Verlag