Exercise · ZW-HAUSDORFF-SPACE-010 · ★★★ Advanced
Let \(\mathbb{N}\) have the cofinite topology. Which statement is correct?
Choose one option.
Show answer
- B. Every singleton is closed, but it is not Hausdorff
Explanation
Each \(\{n\}\) is finite, hence closed. But two nonempty open sets are complements of finite sets and always meet, so nothing separates points. Closed singletons are necessary, not sufficient.
Review the concept: Hausdorff space
A topological space \(X\) is Hausdorff if for every pair of distinct points \(x\) and \(y\) there exist disjoint open sets \(U\ni x\) and \(V\ni y\).
\[x\neq y\ \Rightarrow\ \exists\,U,V\ \text{open}:\ x\in U,\ y\in V,\ U\cap V=\varnothing\]
More practice problems: hausdorff space
- ZW-HAUSDORFF-SPACE-001 Which of these topologies on X=\1,2,3\ is Hausdorff?
- ZW-HAUSDORFF-SPACE-002 How many of the 29 possible topologies on a 3-point set are Hausdorff?
- ZW-HAUSDORFF-SPACE-003 Which of these topologies on ℝ is not Hausdorff?
- ZW-HAUSDORFF-SPACE-004 In X=\1,2,3,4\ with open sets ∅, \1\, \2\, \1,2\, \3,4\, \1,3,4\, \2,3,4\, \1,2,3,4\, how many pair…
- ZW-HAUSDORFF-SPACE-005 If X is Hausdorff and A⊆ X, the subspace A is:
- ZW-HAUSDORFF-SPACE-006 In a Hausdorff space X, what can be said about each singleton \x\?
- ZW-HAUSDORFF-SPACE-007 In X=\1,2,3,4\ with open sets ∅, \1\, \4\, \1,4\, \2,3\, \1,2,3\, \2,3,4\, \1,2,3,4\, which pair of…
- ZW-HAUSDORFF-SPACE-008 Let f:X→ Y be continuous and injective, with Y Hausdorff. What follows about X?