This is from Lee's Introduction to Smooth Manifold (2nd Ed.) page 101. What do we mean with a single k-slice? Maybe a connected one?
2026-04-12 00:43:02.1775954582
Definition of single k-slice
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The meaning here is completely literal: "single $k$-slice" means "one $k$-slice". If you like, the word "single" can be omitted, so the statement is just that $S\cap U$ is a $k$-slice of $U$. The word "single" is emphasizing that $S\cap U$ is equal to just one $k$-slice of $U$ rather being a union of multiple different $k$-slices.