We divided this thesis into two larger chapters in which we observed various aspects of connection. In the first chapter of this work, we studied the definitions of norm, metric and equivalent metric in the context of metric spaces. We also explored open and closed sets in these spaces, as well as continuity of functions between them. After that, we introduced the term of topology and topological spaces. Also, we studied the bases of a topology and continuity of functions between topological spaces. Furthermore, we researched subspaces of metric and topological spaces. In the second chapter, we defined the notion of separation in topological and metric spaces, as well as connected topological and metric spaces. We proved that every segment [a,b] is connected. Then we studied the connected components of a node in topological spaces and some of their properties. We also defined the product of topological spaces and proved that the product of two connected topological spaces is connected. We introduced the notion of a path-connected topological space, proved that every path-connected topological space is connected topological space, and gave an example of a topological space that is connected but not path-connected. Furthermore, we investigated local connectivity and provided an example of a topological space that is connected but not locally connected. Finally, we proved Baire’s category theorem and using it the theorem that claims that a connected, locally connected, and complete metric space cannot be represented as the union of countably many closed, real, and mutually disjoint subsets.