Abstract:
By analogy with the spaces $C_p(X)$, we introduce into consideration the spaces $C_{p,A}(X)$ of continuous realvalued functions with the topology of pointwise convergence on an everywhere dense subset $A \subset X$. For the spaces $C_{p,A} [a,b]$ and $C_{p,B} [c,d]$, where $A=[a,b]\setminus\{a_1,a_2,\dots. a_n\}$, and $B = [c,d]\setminus\{b_1,b_2,\dots.b_m\}$, it is proved that the spaces $C_{p,A} [a,b]$ and $C_{p,B} [c,d]$ are linearly homeomorphic if and only if $n = m$.
Keywords:continuous functions, homeomorphism, pointwise convergence topology, linearly bounded functionals, function of bounded variation, Stieltjes integral.
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