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Object extraction from either aerial or satellite images is a research focus in the field of image processing, where it is essential to determine whether pixels on an image should be classified as object or background. This is a difficult issue, particularly for those pixels which are uncertain for such a determination. This paper presents a newly developed object extraction method that is based on fuzzy topology. First, the idea that the concept of object extraction is understood as a fuzzy set in fuzzy topological space is introduced, i.e., the fuzzy object in object extraction, which is composed of its interior and a portion of its boundary, in fuzzy topological space is introduced. Second, the segmentation of the fuzzy object into the interior and the boundary is described. Two threshold values induced by the optimal threshold value are proposed for determining the interior, boundary, and exterior of the fuzzy object. Third, object extraction regarding the boundary is introduced, where the connectivity analysis is applied to determine whether the pixels within the boundary should be classified as the object. Finally, as an example of application, the developed fuzzy-topology-based object extraction method is applied to boulder extraction from aerial images. The experimental results demonstrate that the newly proposed method provides a better object extraction result, particularly to those boundary pixels which are difficult to determine if they should be classified as the object by the existing solutions. The main contribution of this paper includes the following two points: 1) Theoretically, this paper introduces fuzzy topology into object extraction where the object and the image background are understood as fuzzy sets in fuzzy topological space, and 2) this paper newly proposes the connectivity analysis method for determining if the pixels within the object boundary should be classified as the object. The overall quality of object extraction is thus improved - - greatly.