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Math and science::Topology

Metric space. Open and closed sets

Open and closed subsets

Let \( X \) be a metric space.

Open in \( X \)
A subset \( U \) of \( X \) is open in \( X \) iff [...].
Closed in \( X \)
A subset \( V \) of \( X \) is closed in \( X \) iff [...].

Tom Leinster describes the openness of \( U \):

Thus, \( U \) is open if every point of \( U \) has some elbow room—it can move a little bit in each direction without leaving \( U \).

Personally, I like the phrase: every element of an open set has a neighbourhood.


Open ε-balls are open, and closed ε-balls are closed. Consider trying to prove this. They are open on account of the definition of openness, not by their own definition alone, despite their names being suggestive.