The Standard Topology on Real Numbers

Topology · Axiom Academy

EXAMPLE Standard Topology on ℝ Three worked examples exploring open sets, bases, and closedness Problem 1: Open Intervals Form a Basis To prove this, we need to verify the two conditions for a basis: We'll show that for every point in (0,1), we can find an open ball contained in (0,1): We'll use the definition: a set is closed if and only if its complement is open: A basis for a topology must cover the entire space and have the property that intersections of basis elements can be represented as unions of basis elements To prove a set is open, show that every point has an open ball (or basis element) around it contained in the set To prove a set is closed, it's often easier to show its complement is open The standard topology on ℝ uses open intervals as its basis, making "open in the topology" equivalent to "union of open intervals"

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