  D. K. Biss (Topology and its Applications 124 (2002) 355-371) introduced the
topological fundamental group and presented some interesting basic properties
of the notion. In this article we intend to extend the above notion to homotopy
groups and try to prove some similar basic properties of the topological
homotopy groups. We also study more on the topology of the topological homotopy
groups in order to find necessary and sufficient conditions for which the
topology is discrete. Moreover, we show that studying topological homotopy
groups may be more useful than topological fundamental groups.
