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In real-world environments, a well-designed model must be capable of handling dynamically evolving distributions, where both in-distribution (ID) and out-of-distribution (OOD) samples appear unpredictably and individually, making real-time adaptation particularly challenging. While open-set test-time adaptation has demonstrated effectiveness in adjusting to distribution shifts, existing methods often rely on batch processing and struggle to manage single-sample data stream in open-set environments. To address this limitation, we propose Open-IRT, a novel open-set Intermediate-Representation-based Test-time adaptation framework tailored for single-image test-time adaptation with vision-language models. Open-IRT comprises two key modules designed for dynamic, single-sample adaptation in open-set scenarios. The first is Polarity-aware Prompt-based OOD Filter module, which fully constructs the ID-OOD distribution, considering both the absolute semantic alignment and relative semantic polarity. The second module, Intermediate Domain-based Test-time Adaptation module, constructs an intermediate domain and indirectly decomposes the ID-OOD distributional discrepancy to refine the separation boundary during the test-time. Extensive experiments on a range of domain adaptation benchmarks demonstrate the superiority of Open-IRT. Compared to previous state-of-the-art methods, it achieves significant improvements on representative benchmarks, such as CIFAR-100C and SVHN — with gains of +8.45\% in accuracy, -10.80\% in FPR95, and +11.04\% in AUROC.