Metric spaces · Definition

Open set in a metric space

topologymetric spaces

Before you start you need to know

Open sets are those with no boundary of their own: from any of their points you can move a little in every direction without leaving. All of topology is built from them.

Definition

A subset \(U\) of a metric space \((X,d)\) is open if every point \(x\in U\) is the center of some open ball \(B(x,r)\) contained in \(U\).

\[\forall x\in U\ \ \exists r>0:\ B(x,r)\subseteq U\]

In usual \(\mathbb{R}\), intervals \((a,b)\) are open and \([a,b]\) are not, because at an endpoint every ball sticks out. In \(\mathbb{R}^2\), a disk without its circle is open; with the circle it is not. The empty set and the whole space \(X\) are always open (the empty set meets the condition with no points to check).

Every open ball is an open set: if \(y\in B(x,r)\) and \(s=r-d(x,y)>0\), the triangle inequality gives \(B(y,s)\subseteq B(x,r)\). That is why balls are the basic building blocks: a set is open exactly when it is a union of open balls.

Arbitrary unions of open sets are open, and so are finite intersections. The word «finite» matters: \(\bigcap_n(-1/n,1/n)=\{0\}\) is not open. These three properties (plus \(\varnothing\) and \(X\)) are the ones later taken as the axioms of a topology.

Being open depends on the ambient space: \((1/2,1]\) is open in \(X=(0,1]\) but not in \(\mathbb{R}\). With the discrete metric every subset is open, and replacing the metric by an equivalent one (Euclidean, taxi or max) does not change which subsets of \(\mathbb{R}^2\) are open.

Think about it

If \(U\) is open and \(x\in U\), can one radius \(r\) work for all points of \(U\) at once?

In general no. In \(U=(0,1)\), points near \(0\) or \(1\) require ever smaller radii; the radius depends on the point.

Is \(\mathbb{R}\) (the \(x\)-axis) open as a subset of \(\mathbb{R}^2\)?

No. Any disk centered at a point of the axis contains points with \(y\ne0\), so no ball fits inside the axis.

Exercises 10

  1. In usual \(\mathbb{R}\), which of these sets is open?
  2. Let \(U=(0,3)\) in \(\mathbb{R}\). What is the largest radius \(r\) with \(B(1,r)\subseteq U\)?
  3. Why is \((0,1]\) not open in usual \(\mathbb{R}\)?
  4. In \(\mathbb{R}^2\) with the Euclidean distance, which of these sets is open?
  5. In a metric space \((X,d)\) with the discrete metric, which subsets are open?
  6. Let \(U=\{x\in\mathbb{R}: x^2<4\}\) with the usual distance. Is it open?
  7. What is \(\bigcap_{n\ge1}\left(-\tfrac1n,\tfrac1n\right)\) in \(\mathbb{R}\)?
  8. Let \(U=B(0,1)\) in \(\mathbb{R}\). What is the largest radius \(r\) with \(B(0.9,r)\subseteq U\)?
  9. Is \(\mathbb{Q}\) an open subset of usual \(\mathbb{R}\)?
  10. Let \(X=(0,1]\) with the usual distance. Is \((1/2,1]\) open in the space \(X\)?

What comes next

References

  • James R. Munkres, Topology, 2nd edition, Pearson, 2000. §20
  • Bert Mendelson, Introduction to Topology, 3rd edition, Dover Publications, 1990. cap. 2