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3 added 999 characters in body

Yes, there are plenty of such things.

[In the following, "compact" implies "locally compact" implies "Hausdorff".]

1) To a Boolean algebra, one associates its Stone space, a compact totally disconnected space.

(Via the correspondence between Boolean algebras and Boolean rings, this is a special case of the Zariski topology -- but with a distinctive flavor -- that predates it.)

2) To a non-unital Boolean ring one associates its Stone space, a locally compact totally disconnected space.

3) To a commutative C*-algebra with unit, one associates its Gelfand spectrum, a compact space.

4) To a commutative C*-algebra without unit, one associates its Gelfand spectrum, a locally compact space.

6) To a commutative Banach ring [or a scheme over a non-Archimedean field, or...] one associates its Berkovich analytic space spectrum (the bounded multiplicative seminorms).

7) To a commutative ring R, one associates its real spectrum (prime ideals, plus orderings on the residue domain.)

8) To a field extension K/k, one associates its Zariski Riemann surface (equivalence classes of valuations on K which are trivial on k).

This is by no means a complete list...

Addendum: I hadn't addressed the second part of your question, i.e., explaining what these things are used for. Briefly, the analogy to the Zariski spectrum of a commutative ring is tight enough to give the correct impression of the usefulness of these other spectra/spaces: they are topological spaces associated (cofunctorially) to the algebraic (or algebraic-geometric, topological algebraic, etc.) objects in question. They carry enough information to be useful in the study of the algebraic objects themselves (sometimes, e.g. in the case of Stone and Gelfand spaces, they give complete information, i.e., an anti-equivalence of categories, but not always). In some further cases, one can get the anti-equivalence by adding further structure in a very familiar way: one can attach structure sheaves to these guys and thus get a class of "model spaces" for a certain species of locally ringed spaces -- e.g., Berkovich spectra glue together to give Berkovich analytic spaces.

2 deleted 35 characters in body

Yes, there are plenty of such things.

[In the following, "compact" implies "Hausdorff". Yes, I an arithmetic/algebraic geometer.]locally compact" implies "Hausdorff".]

1) To a Boolean algebra, one associates its Stone space, a compact totally disconnected space.

(Via the correspondence between Boolean algebras and Boolean rings, this is a special case of the Zariski topology -- but with a distinctive flavor -- that predates it.)

2) To a non-unital Boolean ring one associates its Stone space, a locally compact totally disconnected space.

3) To a commutative C*-algebra with unit, one associates its Gelfand spectrum, a compact Hausdorff space.

4) To a commutative C*-algebra without unit, one associates its Gelfand spectrum, a locally compact Hausdorff space.

6) To a commutative Banach ring [or a scheme over a non-Archimedean field, or...] one associates its Berkovich analytic space (bounded multiplicative seminorms).

7) To a commutative ring R, one associates its real spectrum (prime ideals, plus orderings on the residue domain.)

8) To a field extension K/k, one associates its Zariski Riemann surface (equivalence classes of valuations on K which are trivial on k).

This is by no means a complete list...

1

Yes, there are plenty of such things.

[In the following, "compact" implies "Hausdorff". Yes, I an arithmetic/algebraic geometer.]

1) To a Boolean algebra, one associates its Stone space, a compact totally disconnected space.

(Via the correspondence between Boolean algebras and Boolean rings, this is a special case of the Zariski topology -- but with a distinctive flavor -- that predates it.)

2) To a non-unital Boolean ring one associates its Stone space, a locally compact totally disconnected space.

3) To a commutative C*-algebra with unit, one associates its Gelfand spectrum, a compact Hausdorff space.

4) To a commutative C*-algebra without unit, one associates its Gelfand spectrum, a locally compact Hausdorff space.

6) To a commutative Banach ring [or a scheme over a non-Archimedean field, or...] one associates its Berkovich analytic space (bounded multiplicative seminorms).

7) To a commutative ring R, one associates its real spectrum (prime ideals, plus orderings on the residue domain.)

8) To a field extension K/k, one associates its Zariski Riemann surface (equivalence classes of valuations on K which are trivial on k).

This is by no means a complete list...