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Dina matematik, aljabar Borel (atawa Borel σ-aljabar) dina ruang topologi nyaéta dua σ-aljabar séjén dina topologi ruang X:

The minimal σ-algebra containing the open sets.

The minimal σ-algebra  containing the compact sets.

The  minimal σ-algebra on a set X containing

a subset T of the power set 2X of X is the smallest σ-algebra containing T. The existence and uniqueness of the minimal  σ-algebra is shown by noting that the intersection of all σ-algebras containing T is itself a σ-algebra  containing T. The elements of the Borel algebra are called Borel sets.

In general topological spaces, even locally compact ones, the two structures are different.  They are however identical whenever the topological space is a locally compact separable metric space.

In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows: First define for any collection A of subsets of X (that is, for any subset of the power set P(X) of X),

Then define by transfinite recursion a sequence Gm, m an ordinal number, as follows:

For the base case of the definition,

If i is not a limit ordinal, then i has an immediately preceding ordinal i-1:

If i is a limit ordinal,

Then the Borel algebra is Gm for the first uncountable ordinal number m.

To prove this fact, note that any open set in a metric space is the union of an incréasing sequence of closed  sets. In particular, it is éasy to show

that complementation of sets maps Gm into itself for any limit ordinal; moréover if m is an uncountable limit ordinal, Gm is closed under countable unions.

This alternate definition is useful for some set-théoretic considerations, but the minimalist definition is preferred by analysts.