Write a PREreview

We develop foundational elements of a topology whose open sets are single-valued (T,I,N,F)-neutrosophic sets. The four coordinates represent truth, pure indeterminacy, neutrality, and falsehood, and the abbreviation TINF is used only for notational convenience. After fixing the order and complement needed to define inclusion, union, intersection, and duality, we introduce single-valued (T,I,N,F)-neutrosophic topological spaces and study closed sets, interior, closure, boundary, subspaces, products, and continuous maps. Classical topologies are represented naturally through crisp (T,I,N,F) characteristic sets, yielding an ordinary crisp core for every TINF topology. We also compare the resulting topological structure with related four-component theories: a coordinate permutation identifies it with quadripartitioned neutrosophic topology, while another identifies the underlying min--max open-set structure with the Turiyam value structure, although the published Fuzzy Neutrosophic Turiyam complement is not preserved. The main topological construction is the space of TINF points. We define the ordinary topology induced on this point space and prove that TINF continuity is equivalent to ordinary continuity of the induced point map. We then establish exact support-based correspondences for the separation axioms $T_0$, $T_1$, and $T_2$, and investigate connectedness and compactness. Although $V_{A\wedge B}=V_A\cap V_B$, the inclusion $\bigcup_\lambda V_{A_\lambda}\subseteq V_{\bigvee_\lambda A_\lambda}$ may be strict; this leads to strengthened notions of TINF connectedness and compactness that correspond exactly to ordinary connectedness and compactness of the induced point space.