| Summary: | In this paper, we originate a new class of open sets, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets, and we review its important properties. Then, some separation axioms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets are introduced and investigated. In addition, we define the so-called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-compact and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="monospace">C</mi><mo>′</mo></msup></semantics></math></inline-formula>-compact spaces via <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets, and the theorems based on them are discussed with counterexamples. Moreover, we entitle the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-continuous and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="monospace">C</mi><mo>′</mo></msup></semantics></math></inline-formula>-continuous functions by applying <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="monospace">C</mi></semantics></math></inline-formula>-open sets. In particular, several inferred properties of them and their connection with the other topological spaces are studied theoretically. Many examples are given to explain the concepts lucidly. The results established in this research paper are new in the field of topology.
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