Final answer:
Apache Hadoop is an open-source framework for distributed processing and analysis of big data sets in clusters, featuring storage via HDFS and processing through MapReduce.
Step-by-step explanation:
The open-source framework used for distributed processing and analysis of big data sets in clusters is Apache Hadoop. Apache Hadoop is designed to scale up from single servers to thousands of machines, each offering local computation and storage. Its core components are Hadoop Distributed File System (HDFS) for storage, and MapReduce for processing. While the other options listed, such as Apache Spark, Apache Storm, and Apache Flink, are also used for processing big data, they are primarily known for their speed, streaming capabilities, and in-memory computation rather than the foundational distributed processing and storage capabilities that Hadoop provides.