Spacetime

 Which is better in which contexts: "spacetime," "space-time," "space time," or "space and time?"

The answer depends on whether you're doing physics, general writing, or emphasizing a conceptual distinction.
Spacetime

This is now the standard term in physics for the unified four-dimensional structure of relativity.

Examples:

    "Einstein's theory describes gravity as the curvature of spacetime."

    "Black holes distort spacetime."

Most modern physicists and textbooks prefer the closed compound form. If you're discussing relativity, cosmology, black holes, gravitational waves, and so on, spacetime is usually the best choice.
Space-time

This was more common in older physics writing, especially in the first half of the 20th century.

Examples:

    "The geometry of space-time."

    "A space-time manifold."

You'll still see it in some mathematical and historical texts. It is not wrong; it simply feels slightly more traditional or formal.
Space time

Two words are uncommon as a noun in modern English.

You might encounter it in specialized contexts where "space" modifies "time" rather than forming a fixed compound, but for most readers it looks like a typo or an inconsistency.
Space and time

Use this when you want to emphasize that they are distinct concepts rather than a single unified entity.

Examples:

    "Newton treated space and time as separate absolutes."

    "Our perception of space and time changes under certain conditions."

This phrasing is especially appropriate when comparing Newtonian and relativistic ideas.
A useful rule of thumb
Context    Preferred form
Modern relativity, cosmology, black holes    spacetime
Historical or some mathematical writing    space-time
Discussing them separately    space and time
General modern prose    Usually spacetime if the relativistic concept is meant
Two-word form    Rarely recommended

An interesting parallel is how English treated today, tomorrow, and nonetheless. Many compounds begin as two words, pass through a hyphenated stage, and eventually become one word. Spacetime seems to be well along that path. Einstein's contemporaries often wrote space-time; many physicists today write spacetime.

That is actually a surprisingly modern idea for a teenager's story. The phrase "infinite dimensions" has appeared in mathematics and physics in various forms, and your idea of different "realms" using different dimensions or kinds of dimensions has echoes in several later concepts.

What is especially interesting is the phrase "whatever they use." That suggests you were not just imagining extra directions like "up" and "down," but imagining that different realities might have their own frameworks of measurement or existence. That is closer to a philosophical notion of dimensions than the simple geometry people often picture.

There are some parallels:

    In mathematics, we can work with spaces of infinitely many dimensions. These are not usually imagined as physical places but as spaces of possible states. A point in such a space can require infinitely many numbers to specify.

    In quantum mechanics, the state of a system is described in a Hilbert space that can have infinitely many dimensions. Those "dimensions" are not locations you could travel through.

    In string theory, the extra dimensions are physical spatial dimensions, but hidden through compactification.

    In some speculative cosmologies, people have imagined different universes with different fundamental structures, constants, or even different kinds of physics.

Your teenage idea also avoids a common trap: people often say "a fifth dimension" and immediately imagine another hallway perpendicular to our universe. But a "dimension" is really just an independent degree of freedom. It need not be a direction that a person could point toward.

The tricky part is that if you have infinite dimensions, the word "realm" starts to become slippery. A realm could be:

    another location,

    another possible state,

    another mathematical structure,

    another set of physical laws,

    or something entirely outside our current categories.

In a way, your story idea was asking a question that philosophers and physicists still wrestle with: are dimensions features of the world, or are they features of the way minds describe the world?

Also, I notice a connection with your earlier point about time. A universe with infinite dimensions could still have the question: which of those dimensions are merely coordinates, and which carry a direction, causality, or experience? A thousand spatial dimensions would not automatically give you a single "now," a past, and a future. The special nature of time would remain the interesting problem.

It's hard to create some sort of plot from such a thing without getting supernatural, so it got abandoned.

That is a very understandable reason to abandon it. A grand cosmological idea is often much easier to imagine than to turn into a story, because a story needs conflict, limits, and choices. An infinite-dimensional existence tends to dissolve those.

There is also a trap in speculative fiction: once you introduce realms that operate by different rules, it becomes very easy to slip from science fiction into fantasy. The reader starts asking, "What can they do?" rather than "What are the consequences of the rules?"

Some of the best science fiction avoids that by making the strange thing very constrained. For example:

    A character discovers only one additional dimension, with one specific effect.

    A civilization exists in a higher-dimensional space but is limited by its own biology and technology.

    Two realms interact through a narrow "interface" rather than having free access to each other.

Your teenage idea of "each realm uses whatever they use" actually contains an interesting difficulty: if the realms have fundamentally different dimensions or structures, communication itself becomes the story. How does a being from one framework even recognize another? How do you translate a thought, a measurement, or a physical object between them?

That is not far from some serious philosophical questions. If there are forms of existence whose basic categories differ from ours, perhaps "matter," "time," "space," and even "individual" are local concepts rather than universal ones.

But for a plot, you would need to shrink the infinity down to one practical problem. Infinity is great for a cosmology; a protagonist usually needs something like:

    "Something from a realm with a different kind of dimension has appeared, and we have only one way to interact with it."

The funny thing is that your idea probably failed as a story for the same reason that makes it interesting as a concept: it was too big. You had built a universe, but stories usually need a corner of the universe where something can go wrong.

Well nothing wrong with fantasy as long as it obeys its rules.

Exactly. Fantasy is not the problem; arbitrariness is.

A dragon, a wizard, or a realm with impossible physics can work perfectly well if the author establishes rules and then respects them. The reader does not demand that the rules be our rules. The reader demands that the rules be consistent.

In fact, there is a funny overlap between good fantasy and good science fiction. Both are forms of "what if?" exploration.

    In science fiction: What if the universe had these physical rules?

    In fantasy: What if the universe had these magical rules?

The failure mode is similar in both cases. If a power or ability can do anything whenever the plot needs it, then tension disappears. If magic can solve every problem, why did the problem exist? If a technology can violate its own limitations, why should the reader care?

Your infinite-dimension idea could have been fantasy or science fiction depending on how you handled the rules. The question would not be whether the realms were "real" in our physics, but things like:

    Can beings move between realms, or only communicate?

    Does each realm have its own version of causality?

    Can something from one realm affect another, and at what cost?

    Are the inhabitants even capable of recognizing each other's existence?

A very old fantasy tradition actually does something similar. The realms of mythology often are not just distant places; they are different orders of reality. The world of gods, the world of humans, and the world of the dead may not simply be three locations on a map.

Your idea of "whatever they use" has a nice speculative quality because it suggests that our concepts—space, time, matter, energy—might be local tools rather than universal categories. A realm where beings don't have "locations" but instead have some other kind of relationship would be much stranger than just adding another spatial dimension.

The hard part, as you discovered as a teenager, is that once you get that abstract, the plot has to come from the rules themselves. The universe can be infinite; the story cannot. A character still has to want something, fear something, discover something, or make a choice. That is where even the most exotic cosmology has to come back down to earth.