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While class activation map (CAM) generated by image classification network has been widely used for weakly supervised object localization (WSOL) and semantic segmentation (WSSS), such classifiers usually focus on discriminative object regions. In this paper, we propose Contrastive learning for Class-agnostic Activation Map (C$^2$AM) generation only using unlabeled image data, without the involvement of image-level supervision. The core idea comes from the observation that i) semantic information of foreground objects usually differs from their backgrounds; ii) foreground objects with similar appearance or background with similar color/texture have similar representations in the feature space. We form the positive and negative pairs based on the above relations and force the network to disentangle foreground and background with a class-agnostic activation map using a novel contrastive loss. As the network is guided to discriminate cross-image foreground-background, the class-agnostic activation maps learned by our approach generate more complete object regions. We successfully extracted from C$^2$AM class-agnostic object bounding boxes for object localization and background cues to refine CAM generated by classification network for semantic segmentation. Extensive experiments on CUB-200-2011, ImageNet-1K, and PASCAL VOC2012 datasets show that both WSOL and WSSS can benefit from the proposed C$^2$AM.