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In this paper it is proved that if      C 0 ( X ) {C_0}(X)       and      C 0 ( Y ) {C_0}(Y)       are topologically isomorphic then      | X | = | Y | |X| = |Y|      .

On topological isomorphisms of 𝐶₀(𝑋) and the cardinal number of 𝑋