Trivial

Meaning

Trivial means "obvious", if something is trivial it means very little work needs to be done to show it holds. Trivial can also mean the argument is vacuous, for example [ilmath]\forall x\in X[P(X)][/ilmath] holds, when [ilmath]X\eq\emptyset[/ilmath] is it vacuously true.

Disambiguation

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Trivial may refer to:

• Algebraic objects:
  • Trivial (group) - the group: [ilmath](\{e\},*:\{e\}\times\{e\}\rightarrow\{e\})[/ilmath] is called the trivial group, [ilmath]e[/ilmath] is the identity element and [ilmath]e*e\eq e[/ilmath] is the only operation
  • Trivial (ring)
  • Trivial (module)
  • Trivial (vector space)
• Spaces:
  • Trivial (ring of sets)
  • Trivial (sigma-ring of sets)
  • Trivial (topology) - AKA: indiscrete topology
    • Caveat:The trivial metric induces basically the exact opposite topology, the trivial metric, [ilmath]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x\eq y\\ 1&\text{otherwise}\end{array}\right.[/ilmath], induces the discrete topology - the power-set topology.
  • Trivial (metric)