The Definition and Basic Properties of Topological Groups
نویسنده
چکیده
For simplicity, we follow the rules: S denotes a 1-sorted structure, R denotes a non empty 1-sorted structure, X denotes a subset of the carrier of R, T denotes a non empty topological structure, and x denotes a set. Let X, Y be sets. One can verify that every function from X into Y which is bijective is also one-to-one and onto and every function from X into Y which is one-to-one and onto is also bijective. Let X be a set. Observe that there exists a function from X into X which is one-to-one and onto. Next we state the proposition (1) rng(idS) = ΩS . Let R be a non empty 1-sorted structure. Note that (idR) −1 is one-to-one. We now state two propositions: (2) (idR) −1 = idR. (3) (idR) (X) = X. Let S be a 1-sorted structure. One can check that there exists a map from S into S which is one-to-one and onto.
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