D. K. Biss (Topology and its Applications 124 (2002) [355][356][357][358][359][360][361][362][363][364][365][366][367][368][369][370][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.2000 Mathematics Subject Classification. 55Q05; 55U40; 54H11; 55P35.

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