## 1 Introduction  
Topological insulators (TIs) have an insulating gap in the bulk and gapless surface states at the same time, and these insulators have been extensively studied in past decades [1, 2] due to their potential applications in spintronic devices [3, 4]. Therefore, TIs have attracted much attention in both the fundamental physics and experimental research fields. Studies on BiSe\({}_{2}\)[5], half-Heusler alloys such as LnPtBi\({}_{3}\)[6, 7], and binary compounds such as HgTe [8, 9] have demonstrated that those materials are normal insulators, but they may also be transformed into TIs by applying spin-orbit coupling (SOC), axial or uniaxial strain, and impurities. Topologically trivial to nontrivial transitions of TIs are identified by observing the band inversion in the Brillouin zone. [PERSON] and [PERSON] [10] introduced a useful and practical Chern number or topological number \(Z_{2}\), which is calculated by analyzing the bulk band structure and provides a physical meaning in the prediction of edge states at interfaces.  
Perovskite crystals are of interest in the fields of high thermoelectric power, ferroelectricity, superconductivity, spin electronic devices, and colossal magnetoresistance [11, 12, 13]. Studies on perovskites have been aimed at, for example, energy-saving and cost efficiency as well as the manufacture of stable solar cells [14]. For example, the cesium lead halide perovskite (CsPbX\({}_{3}\); \(X\) = Cl, Br, and I) has received considerable attention for its potential application in high-performance solar cells [15; 16] and light-emitting diodes [17; 18]. Furthermore, organic lead halide (CH\({}_{3}\)NH\({}_{3}\)Pb\(X_{3}\)) nanocrystals and bulk crystals are promising high-efficiency solar cells with power conversion efficiencies exceeding 20% [19]. The TI phase in perovskites and double perovskites has been studied theoretically [20; 21; 22]. The coexistence of the TI phase and other phases, such as those of superconductivity, magnetism, and ferroelectricity, are of particular interest. [PERSON] et al. [23] reported that cesium lead halide perovskite (CsPbI\({}_{3}\)) has a TI phase under strain together with a ferroelectric phase.  
Previous theoretical studies on CsPb\(X_{3}\) (\(X\) = Cl, Br, and I) have focused on its electronic [23; 24; 25; 26] and optical [24; 25] properties. For example, [PERSON] et al. [25] reported that CsPbI\({}_{3}\) becomes a TI under very high pressure [23; 26]. However, the reported band gap for the CsPbI\({}_{3}\) in the fully relativistic scheme is significantly underestimated. The calculated band gap of CsPbI\({}_{3}\) is 1.48 eV without SOC in the generalized gradient approximation (GGA) and becomes 0.29 eV when the SOC is considered, revealing that the SOC effect is strong in these cesium lead halide perovskites. Nonetheless, the optical band gap for CsPbI\({}_{3}\) has been reported to be 1.73 eV experimentally [27; 28].  
In this study, we investigated the TI phase of CsPbBr\({}_{x}\)I\({}_{3-x}\) (\(x\) = 0, 1, 2, and 3) under hydrostatic strain. The paper is organized into the following sections: Section 2 describes the crystal structures and computational methods, initially addressing the structural and electronic properties with and without consideration of SOC. Section 3 depicts the bulk and surface band structures and describes the calculation of the electric polarization produced by Pb atom distortion. We confirm theoretically that the slight distortion will not disrupt the TI phase of cesium lead halide perovskite. In the final section, we draw conclusions.