Real Projective Line - Intervals and Topology

Intervals and Topology

The concept of an interval can be extended to . However, since it is an unordered set, the interval has a slightly different meaning. The definitions for closed intervals are as follows (it is assumed that ):

\begin{align} \left & = \lbrace a \rbrace \\ \left & = \lbrace x \vert x \in \mathbb{R}, a \leq x \leq b \rbrace \\ \left & = \lbrace x \vert x \in \mathbb{R}, a \leq x \rbrace \cup \lbrace \infty \rbrace \\ \left & = \lbrace x \vert x \in \mathbb{R}, b \leq x \rbrace \cup \lbrace \infty \rbrace \cup \lbrace x \vert x \in \mathbb{R}, x \leq a \rbrace \\ \left & = \lbrace \infty \rbrace \cup \lbrace x \vert x \in \mathbb{R}, x \leq a \rbrace \\ \left & = \lbrace \infty \rbrace \end{align}

The corresponding open and half-open intervals are obtained by removing the endpoints.

itself is also an interval, but cannot be represented with this bracket notation.

The open intervals as base define a topology on . Sufficient for a base are the finite open intervals and the intervals .

As said, the topology is homeomorphic to a circle. Thus it is metrizable corresponding (for a given homeomorphism) to the ordinary metric on this circle (either measured straight or along the circle). There is no metric which is an extension of the ordinary metric on R.

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