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Oct 19, 2023 at 16:35 comment added YCor OK, fine. I'd like to know who coined this terminology... Wikipedia quotes Cantor 1883, I doubt it...
Oct 19, 2023 at 16:20 comment added Joseph Van Name The term 'uniformly connected' has already been established, and it is the analogue of connectedness for uniform spaces. A uniform space $X$ is uniformly connected iff the only uniformly continuous functions $f:X\rightarrow\{0,1\}$ are the constant functions.
Oct 19, 2023 at 14:12 comment added YCor The terminology "uniformly connected" might be misleading because it is weaker than connected. Maybe "metrically connected"? Examples of metrically connected spaces are dense subsets of connected spaces.
Oct 18, 2023 at 13:56 history answered Joseph Van Name CC BY-SA 4.0