This thesis deals with various analytic constructions of the Cantor set. In the first chapter, we get acquainted with the Cantor set and construct it on the so-called geometric way and we explain some of its properties (length, cardinality and density in the set \(\mathbb{R}\)). In the second chapter, we deal with the first construction of the Cantor set through the so-called continued fractions. We define continued fractions, we prove the existence of a finite representation of every rational number in the form of a continued fraction and the existence of a development into a continued fraction for every real number, that is, we prove the convergence of infinite continued fractions. Finally, we prove that, for an integer \(k\), \(k\geq2\), the set of all real numbers \(\alpha\), such that \(0\leq \alpha\leq k^{-1 }\) and such that the development of the number \(\alpha\) into a continued fraction does not contain a partial quotient smaller than k, the Cantor set. In the third chapter, we construct the Cantor set as a homeomorphic image of the ring of \(p\)-adic integers \(\mathbb{Z}_p\), \(p\) a prime number. First we define \(p\)-adic numbers and whole \(p\)-adic numbers. We define the \(p\)-adic norm and the \(p\)-adic distance on the field \(\mathbb{Q}\) and prove that the field \(\mathbb{Q}\) with the \(p\)-adic the norm is not complete. We complete the field \(\mathbb{Q}\) with the field of equivalence classes of Cauchy sequences, \(\mathbb{Q}_p \). After that, we show the equivalence of the field \(\mathbb{Q}_p \) as a set of equivalence classes of sequences and infinite \(p\)-adic expansions that converge in the \(p\)-adic norm on \(\mathbb{Q} _p\). Finally, we show the existence of a homeomorphism between the ring of \(p\)-adic integers \(\mathbb{Z}_p\) and the Cantor set in the inherited Euclidean topology.