Data science is a recent focus across many fields such as mathematics, statistics and computer sciences aiming at discovering and understand patterns from data. Topological data analysis (TDA) is a solution to discover the underlying topological structures. ,The first part of this thesis is on the random coverage. We estimate the number of data points to cover an underlying manifold with possibly boundaries or singularities and with possibly non-uniform probability distribution. In general, the expected number to cover the space depends not only on the minimum of the probability measure but also on its decaying rates near the minimum point (characterized by two constants).,The second part is about the thresholds of correct Čech homology on a flat torus. The thresholds are estimated by the critical points of distance function and Morse theory. This thesis examines a special type of critical points and makes an improvement on the upper bound of the thresholds over the existing result.,The last part proposes a method to detect topological periodicity in a time series. Topological periodicity allows distortion of the function's domain. The proposed method is about encoding the topological information of a function by a tree. Comparison between functions counting re-parametrization can then be reduced to comparison between two corresponding trees.