Mathematical environments

Mathematical environments

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Quick start: theorems and references

## Continuous maps

> [!def] Continuity
> A map is continuous if the inverse image of every open set is open.

^def-continuity

> [!thm] Composition
> The composition of two continuous maps is continuous.

^thm-composition

Use [[#^def-continuity]] to prove [[#^thm-composition]].

> [!proof]
> Take inverse images successively.

By default, these display as Definition 1.1 · Continuity and Theorem 1.1 · Composition. References display def 1.1 and thm 1.1 and navigate to their targets. Each environment type has its own counter unless shared counters are enabled.

Environment	Accepted callout types	Numbered
Definition	def, definition	Yes
Theorem	thm, theorem	Yes
Lemma	lem, lemma	Yes
Proposition	prop, proposition, prp	Yes
Corollary	cor, corollary	Yes
Claim	claim, clm	Yes
Example	example, ex, exa, exm	Yes
Axiom	axiom, axm	Yes
Assumption	assumption, asm	Yes
Exercise	exercise, exr	Yes
Conjecture	conjecture, cnj	Yes
Hypothesis	hypothesis, hyp	Yes
Proof	proof, pf	No
Remark	remark, rem, rmk	No
Solution	solution, sol	No

Use > [!thm]- Title for an initially collapsed callout, or > [!thm]+ Title for an expanded one. Add another > for each nesting level. Export expands collapsed content.

• Manual number: > [!thm|A.1] Title.
• No number: > [!thm|*] Title. |- and empty metadata | also suppress numbering.
• Automatic number: omit metadata or use |auto. Automatic counters avoid manual numbers already used in the same scope.

Proofs and remarks

Use a proof callout for an unnumbered proof. If the opening line has no title, Proof occupies its own line and the argument starts below. A hollow square □ appears at the end automatically. A custom title or reference can sit beside the first paragraph; add an empty quoted line after the header to start the argument below it.

> [!proof]
> Let $U$ be open. Since $f$ and $g$ are continuous, both
> $f^{-1}(U)$ and $g^{-1}(f^{-1}(U))$ are open.
>
> Thus the composition $f \circ g$ is continuous.

To prove a previously stated theorem, lemma or claim, put its block reference directly after the marker:

> [!proof][[Chapter one#^claim-a]]
> Here is the argument.

This displays Proof of clm 1.1, with a clickable reference. > [!proof] [[#^thm-a]] works too. The reference follows numbering settings, aliases and book chapter renumbering. Use the block ID of the mathematical environment as the target.

A remark callout has an inline title in a brighter shade of the selected palette's main hue, with ordinary text underneath or beside it. It has no frame, shaded background, or QED square.

> [!remark]
> The argument only uses inverse images of open sets.
> It applies to arbitrary topological spaces.

Both environments use normal note text color for their content. pf is an alias for proof; rem and rmk are aliases for remark. Add a title such as > [!proof] Composition or use the usual +/- folding syntax. They support nested callouts and multi-paragraph content. If a proof starts with a list or display equation, its title occupies a separate line; if it ends with one, the square follows on its own final line. Long proofs can continue across PDF pages, with a single square at the end.

Equation numbering

## An identity

$$
a^2+b^2=c^2
$$

^eq-example

See [[#^eq-example]].

By default, only display equations referenced in the indexed vault receive automatic numbers; unreferenced equations do not consume a number. You can number all display equations or disable automatic numbering. Inline math is not numbered.

Manual \tag{A} values are preserved. Without a manual tag, \notag or \nonumber suppresses automatic numbering. Each display-math block receives at most one automatic number; split it into separate blocks for separate references. Blocks containing multiple manual tags retain the original math without guessing a single reference number.