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Real data is often given as a point cloud, i.e. a finite set of points with pairwise distances between them. An important problem is to detect the topological shape of data – for example, to approximate a point cloud by a low-dimensional non-linear subspace such as an embedded graph or a simplicial complex. Classical clustering methods and principal component analysis work well when data points split into good clusters or lie near linear subspaces of a Euclidean space. Methods from topological data analysis in general metric spaces detect more complicated patterns such as holes and voids that persist for a large interval in a 1-parameter family of shapes associated to a cloud. These features can be visualized in the form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimum spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale.

We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree.

A construction of the Urysohn's universal metric space is given in the context of constructive theory of metric spaces. The space is universal in the sense that every separable metric space isometrically embeds into it. Moreover, every isometry between two finite subspaces extends to total isometry, and this determines the Urysohn space uniquely up to isometric isomorphism.