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If an ability would produce one or more mana of an undefined type, it produces no mana instead.

Some abilities produce mana based on the type of mana another permanent or permanents “could produce.” The type of mana a permanent could produce at any time includes any type of mana that an ability of that permanent would produce if the ability were to resolve at that time, taking into account any applicable replacement effects in any possible order. Ignore whether any costs of the ability could or could not be paid. If that permanent wouldn’t produce any mana under these conditions, or no type of mana can be defined this way, there’s no type of mana it could produce.

To “tap [a permanent] for mana” is to activate a mana ability of that permanent that includes the {T} symbol in its activation cost. See rule 605, “Mana Abilities.”

An ability that triggers whenever a permanent “is tapped for mana” or is tapped for mana of a specified type triggers whenever such a mana ability resolves and produces mana or the specified type of mana.

A replacement effect that applies if a permanent “is tapped for mana” or tapped for mana of a specific type and/or amount modifies the mana production event while such an ability is resolving and producing mana or the specified type and/or amount of mana.

If an object’s enters-the-battlefield triggered ability or replacement effect refers to X, and the spell that became that object as it resolved had a value of X chosen for any of its costs, the value of X for that ability is the same as the value of X for that spell, although the value of X for that permanent is 0. This is an exception to rule 107.3i.

The tap symbol is {T}. The tap symbol in an activation cost means “Tap this permanent.” A permanent that’s already tapped can’t be tapped again to pay the cost. A creature’s activated ability with the tap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

The untap symbol is {Q}. The untap symbol in an activation cost means “Untap this permanent.” A permanent that’s already untapped can’t be untapped again to pay the cost. A creature’s activated ability with the untap symbol in its activation cost can’t be activated unless the creature has been under its controller’s control continuously since their most recent turn began. See rule 302.6.

You can activate Prismatic Geoscope's mana ability if you have no basic land types among lands you control, but it won't produce any mana.

The colors of mana that Prismatic Geoscope can produce aren't related to the specific basic land types among lands you control. For example, if you control a Plains and an Island, you can activate Prismatic Geoscope's ability to add {R}{G}.

To determine the number of basic land types among lands you control, look for the Plains, Island, Swamp, Mountain, and Forest subtypes among the lands you control. Count each subtype that appears one or more times toward the number of basic land types among lands you control. This number can't exceed five.

How many lands you control of a particular basic land type is irrelevant to a domain ability, as long as that number is greater than zero. As far as domain is concerned, ten Forests are the same as one Forest.

Prismatic Geoscope can't produce colorless mana.

Most nonbasic lands don't have basic land types, even if they produce colored mana. For example, Caves of Koilos is neither a Plains nor a Swamp.

Some nonbasic lands do have basic land types. Domain abilities don't count the number of lands you control—they count the number of basic land types among lands you control, even if that means checking the same land twice. For example, if you control a Tundra, a Blood Crypt, and a Forest, you'll have a Plains, Island, Swamp, Mountain, and Forest among the lands you control. Your domain abilities will be maxed out.