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Topological properties are with invariance and take priority over other features, which play an important role in cognition. This paper introduces a new attention selection model called TPA (topological properties-based attention), which adopts topological properties and quaternion. In TPA, using Unit-linking PCNN (Pulse Coupled Neural Network) hole-filter expresses an important topological property (the connectivity) in visual attention selection. Meanwhile, using the quaternion Fourier transform based phase spectrum of an image or a frame in a video obtains the spatio-temporal saliency map, which shows the result of attention selection. Adjusting the weight of a topological channel can change its influence. The experimental results show that TPA reflects the real attention selection more accurately than PQFT (Phase spectrum of Quaternion Fourier Transform).