Topological spaces · Definition

Discrete and trivial topology

topologytopological spacesopen sets

Before you start you need to know

They are the two possible extremes of a topology: the one with all the open sets and the one with the fewest.

Definition

The discrete topology \(\mathcal{T}_{disc}\) on a set \(X\) has the power set \(\mathcal{P}(X)\) as open sets; the trivial one \(\mathcal{T}_{triv}\), only the empty set and \(X\).

\[\mathcal{T}_{disc}=\mathcal{P}(X),\qquad \mathcal{T}_{triv}=\{\varnothing,X\}\]

On any set \(X\), every topology \(\mathcal{T}\) satisfies \(\{\varnothing,X\}\subseteq\mathcal{T}\subseteq\mathcal{P}(X)\). The trivial one, also called indiscrete, is the smallest possible and the discrete one is the largest. If \(X\) has \(n\) points, the discrete topology has \(2^n\) open sets and the trivial one always has 2 (or 1 if \(X=\varnothing\)).

In the discrete topology each point \(x\) is open as a singleton \(\{x\}\), and from this every subset is open, being a union of singletons. In the trivial topology the opposite happens: the only open set containing a point is \(X\), so the topology cannot tell points apart. When \(X\) has two or more points the two topologies differ, and on a three-point set there are 27 others between them.

The discrete topology is the one induced by the discrete metric, \(d(x,y)=1\) if \(x\neq y\), since there \(B(x,1)=\{x\}\). The trivial one, in contrast, comes from no metric when \(X\) has at least two points: every metric separates points with small balls, and that would produce proper open sets.

They serve as test cases for any topological definition: if a property fails in the discrete or the trivial topology, it is usually because a hypothesis was forgotten.

Think about it

In the discrete topology on \(X\), is every singleton open?

Yes, because every subset of \(X\) is open, in particular \(\{x\}\).

Why does the trivial topology on a set of two or more points not come from a metric?

With a metric, the ball \(B(x,r)\) with \(r\) smaller than the distance to another point \(y\) is open and does not contain \(y\), so it would be an open set different from \(\varnothing\) and \(X\), impossible in the trivial topology.

Exercises 10

  1. If \(X\) has \(4\) elements, how many open sets does the discrete topology on \(X\) have?
  2. If \(X\) has \(4\) elements, how many open sets does the trivial topology on \(X\) have?
  3. Let \(X=\{a,b,c\}\). In which topology is the set \(\{b\}\) open?
  4. On \(X=\{1,2,3\}\), which family is the discrete topology?
  5. How many topologies on \(\{1,2,3\}\) are neither the discrete nor the trivial one?
  6. In the discrete topology on \(\{1,2,3\}\), how many open sets contain the point \(1\)?
  7. In the trivial topology on \(\{1,2,3\}\), how many open sets contain the point \(1\)?
  8. Let \(\mathcal{T}=\{\varnothing,X,\{1\},\{1,2\}\}\) on \(X=\{1,2,3\}\). How many sets are open in the discrete topology but not in \(\mathcal{T}\)?
  9. Let \(\mathcal{T}\) be any topology on \(X\). Which relation always holds?
  10. When do the discrete and the trivial topologies on a set \(X\) coincide?

References

  • James R. Munkres, Topology, 2nd edition, Pearson, 2000. §12
  • Bert Mendelson, Introduction to Topology, 3rd edition, Dover Publications, 1990. cap. 3