What is closure? — Mathematics

Definition 6.3 University Mathematics — Year 3 · Chapter 6 — General Topology

For $A \subseteq X$ : the interior $\mathring A$ is the largest open set inside $A$ (union of all of them); the closure $\bar A$ the smallest closed set containing $A$ ; the boundary $\partial A = \bar A \setminus \mathring A$ . $A$ is dense if $\bar A = X$ . One has $x \in \bar A$ iff every neighborhood of $x$ meets $A$ (if some neighborhood misses $A$ , its open core’s complement is a smaller closed set around $A$ ; conversely).

Examples

Example 6.22 (The topologist’s sine curve)

Let $\Gamma = \{(x, \sin\frac1x) : 0 < x \leq 1\}$ and $S = \bar\Gamma = \Gamma \cup (\{0\}\times[-1,1])$ (every point $(0, y)$ , $\abs y \leq 1$ , is a limit of points of $\Gamma$ : solve $\sin\frac1x = y$ near $0$ ). Then $S$ is connected — closure of the connected $\Gamma$ , a continuous image of $(0, 1]$ (Theorem 6.20(3)) — but not path-connected: a path from $(1, \sin 1)$ to $(0,0)$ would have to traverse abscissas $\to 0$ while the ordinate oscillates between $\pm1$ ; Exercise 6.9 makes this rigorous. Connectedness and path-connectedness genuinely differ.

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Examples

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