<p>Multi-layer low Earth orbit (LEO) satellite networks consisting of observation satellites (OSs) and communication satellites (CSs) are becoming critical infrastructure to extend the edge-cloud continuum into space. However, the highly dynamic topology of satellite-ground and inter-satellite links presents a significant challenge. It frequently disrupts connectivity, severely impacting the stability and reliability of edge-cloud network operations. To address these issues, this paper proposes a multi-layer satellite-ground integrated topology control method for the space edge-cloud continuum that incorporates spatial-temporal correlations. Especially, a dynamic trajectory model for satellites based on the six Keplerian orbital elements is established. This model precisely characterizes the behaviors of (OSs) edge nodes and (CSs) space relay/fog nodes, laying the foundation for the division of satellite-ground service domains and the establishment of cross-layer links. Then, a dynamic satellite-ground service domain division model is developed, and the satellite-ground service domain division (SGSDD) algorithm is designed. This algorithm dynamically maps high-mobility satellites to multiple ground cloud domains. Consequently, it achieves precise service binding among edge sensing nodes, relay/fog nodes, and ground cloud resources. Furthermore, a bipartite graph matching model for cross-layer links is formulated, and the cross-layer link topology control (CLLTC) algorithm is designed. This algorithm ensures efficient data migration and task offloading between edge nodes and relay nodes. Simulation results verify that the proposed topology control method ensures topology stability and effectively reduces average routing hops under high-speed satellite mobility. Moreover, it significantly enhances data delivery throughput and service reliability within the continuum. This provides robust architectural support for the seamless deployment of future space-based edge intelligence.</p>