Common Pitfalls

“Arbitrary intersections of open sets are open.” False. Only finite intersections are guaranteed. Counterexample: in $\mathbb{R}$, $\bigcap_{n=1}^\infty (-1/n, 1/n) = \{0\}$, which is not open.

“Closed = not open.” False. A set can be both (clopen), neither, or exactly one. In $\mathbb{R}$, $\emptyset$ and $\mathbb{R}$ are clopen; $[0, 1)$ is neither.

“Compact implies closed and bounded in every topological space.” False. This is specific to $\mathbb{R}^n$ (Heine–Borel). In the cofinite topology on an infinite set, every subset is compact but not every subset is closed.

“Connected implies path-connected.” False. The topologist”s sine curve is connected but not path-connected.

“Continuous bijections are homeomorphisms.” False. The bijection $f : [0, 2\pi) \to S^1$ given by $f(t) = (\cos t, \sin t)$ is continuous and bijective, but its inverse is not continuous — $[0, 2\pi)$ is not compact but $S^1$ is.

“Every metric space is complete.” False. $\mathbb{Q}$ with the usual metric is not complete.

“The closure of the interior equals the interior of the closure.” False as a general principle. In $\mathbb{R}$ with $A = \mathbb{Q} \cap (0, 1)$: $\operatorname{int}(\overline{A}) = (0, 1)$ but $\overline{\operatorname{int}(A)} = \emptyset$.