Format of the database entries

On this page we provide some practical information about the format of data files.

Header

Each entry (data file) starts with the header providing essential information about the space, property or theorem allowing it to be used in the deduction engine. Header starts and ends with the line containing just ---. Usually after the header there is also a blank line in the file. In general, any entry in the header starts with a tag followed by a colon and the value after an obligatory space. The important part is that the identifiers have to exactly match the respective file names. While the address https://topology.pi-base.org/spaces/S4 means the same as https://topology.pi-base.org/spaces/S000004, writing S4 as a space identifier in a database entry will cause an error. A correct identifier always consists of a letter and 6 digits.

List of references

Below the fundamental parameters of the entry, there often are references, introduced by the refs: label. Each citation on the list starts with - and consists of two lines. The name: property is self-explanatory. For books and articles it is recommended to indicate the author(s) and year of publication in parentheses (e.g. General Topology (Engelking, 1989)).

$\pi$-Base accepts the following types of links:

• zb: zbMath identifier. In the header it should be in quotation marks (e.g. zb: "0386.54001").
  • Since zbMath is an open access database it is the recommended form of citation, whenever possible. DOIs are acceptable if they are not linked from zbMath and provide a download of the text.
• doi: DOI identifiers (the part after https://doi.org/)
• mr: review number from MathSciNet. Note that you should remove leading zeros.
• wikipedia: name of a Wikipedia page, e.g. Alexandrov_topology. The corresponding name convention is name: Alexandrov topology on Wikipedia.
• mathse: numeric id of questions/answers on Mathematics StackExchange. If the question itself is directly relevant to what we want to show, we use the numeric id of the question itself and the name field will have the title of the mathse post (name: To be, or not to be?). Otherwise, we can refer to a specific answer to the mathse post. The numeric id of that specific answer can be found be clicking on Share below the answer; and the name: value will have the form Answer to "To be, or not to be?" (note the double quotes).
• mo: MathOverflow questions/answers. The same rules as mathse apply.

There is no general url or href reference tag.

Note that both items for the reference (name and doi/zb/mathse/...) have to be lined up with the same identation.

Links and references in the main text

Sources can be referenced in the text by providing it in double brackets {{type:id}}. Do not put a space after the colon. Do not use quotation marks there either. E.g. {{zb:0386.54001}} is the way to insert a link to Counterexamples in topology in the text. References cited in the main body should appear in the refs list in the header.

In the text you can refer to other spaces/properties/theorems by typing {S25}, {P53} or {T89} (in single backets). Referring to spaces known to $\pi$-Base via links of this form is preferred over typing their names. You can cite also the fact whether a certain space satisfies or does not satisfy a given property by using {S64|P133}.

To add any other link you can use general Markdown notation [link text](https://the.url/address). One common use of it is to reference a $\pi$-Base search to show a chain of implications that pi-base is already aware of. We use a link of the form [(Explore)](...). (See here for an example of this usage.)

You can test the links of all (supported) types in the preview window, accessible from the Advanced tab on the pi-base web site. It does not allow you to test the header (e.g. the refs: section), though.

File organisation

It is quite straightforward and intuitive. Currently most (if not all) new files are created by copying and modifying existing ones. The most common mistake is probably forgetting to update the respective identifier to match the file name.

Properties

Each property (type of traits) has its own file in the properties folder of the repository. The file name is just the property's identifier (uid:) with the .md extension.

---
uid: P000005
name: "$T_3$"
aliases:
  - Regular Hausdorff
  - T3
refs:
  - zb: "1052.54001"
    name: General Topology (Willard)
---

A space which is both {P11} and {P3}. 
Equivalently, a space that is both {P11} and {P1}.

Defined in 14.1 of {{zb:1052.54001}}.

----
#### Meta-properties

- The Kolmogorov quotient $\text{Kol}(X)$ is {P5} iff $X$ is {P11}.
- This property is hereditary.

The meta-properties section is optional. You can find more about it here.

Theorems

Each theorem has its own file in the theorems folder. Theorems do not have names or aliases, but a statement consisting of if: and then: parts. The and: operator is not needed if there is only one hypothesis (see for example theorems/T000012.md).

---
uid: T000456
if:
  and:
    - P000057: true
    - P000078: false
then:
  P000181: true
---

By definition.

The above example is the theorem that countable (P57) and not finite (P78) imply countably infinite (P181).

Spaces

Are represented by subfolders of the spaces directory. Each folder's name has to equal the space's id. The description of each space is provided in a file named README.md with a format similar to property definitions. For spaces appearing in Counterexamples in topology there can be an optional tag counterexamples_id:.

Traits

Information whether a certain property is true or false for a given space is provided in a file stored in the subfolder properties of the space's folder. The entry deciding property P3 for space S99 has to be stored in the file spaces/S000099/properties/P000003.md. The parameters space: and property: have to exactly match the ids of the space and of the property, respectively. The latter also coincides with the file name (without extension).

---
space: S000064
property: P000026
value: true
refs:
- zb: "0386.54001"
  name: Counterexamples in Topology
---

$D$ is countable and dense.

See item #3 for space #72 in {{zb:0386.54001}}.