Conventions and Style

Notation

In property descriptions and in proofs of theorems, we often use $X$ to implicitly denote the space being discussed.

When providing the proof of a theorem, for example A + B implies C, we usually make the implicit assumption that the hypotheses, A and B in the example, hold for the space $X$. So there is usually no need to explicitly state it up front.

Separation axioms

For the separation axioms, T_n ⇒ T_m whenever n ≥ m. For example regular is defined to assert that closed points and sets can be separated; T₃ is defined to be both regular and T₀. See e.g. wikipedia for more information.

Local Properties

A property is "locally P" if every point in the space has a neighborhood base satisfying P for every member of the base. On the other hand, some authors define "locally P" to mean there is a single neighborhood satisfying P for each point. These definitions are occasionally equivalent (e.g. locally metrizable), but are not equivalent in general (e.g. locally compact). See this issue for discussion.

Use "locally P" when the P neighborhoods form a neighborhood base, and (when not equivalent) use "weakly locally P" when only a single P neighborhood is required.

Meta-properties

At this moment $\pi$-Base does not support meta-properties but we collect them on property pages for the use in proofs and also for the visitors as additional mathematical facts. Below you can find the list of currently tracked meta-properties and their standardised phrasing:

1. This property is hereditary.

2. This property is hereditary with respect to open sets.

3. This property is hereditary with respect to closed sets.

4. $X$ satisfies this property iff its Kolmogorov quotient $\mathrm{Kol}(X)$ does.

5. This property is preserved by quotient maps.

6. This property is preserved by arbitrary disjoint unions.

7. This property is preserved by $\Sigma$-products

8. This property is preserved by countable products.

9. This property is preserved by arbitrary products.

10. This property is preserved in any finer topology.

11. This property is preserved in any coarser topology.

12. If each point has a neighborhood with the property, $X$ also has the property.

13. If $X$ is covered by countably many subspaces, each having the property, then so does $X$.

14. If each path component of $X$ has the property, then so does $X$.

If the justification is non-trivial, it is good to provide a reference (in parentheses after the meta-property). If the justification is straightforward, there is no need to provide one.

On some occasions analogous negative statements (e.g. This property is not hereditary with respect to open sets) may be of interest. Of course whenever a property is said to be hereditary with respect to e.g. open sets, it usually means it is not hereditary in general.

Note that the list may not be complete; contributors are invited to expand it.

See also the thread #1071.

Notation advice

• Whenever dealing with intervals and pairs (tuples), the latter should be better denoted by $\langle x, y\rangle$ (\langle and rangle) than $(x,y)$. If there is no need to care about the intervals, standard pair notation can be used.
• For open covers the \mathscr font is commonly used.

References

When referencing the literature we shall use zbMath links whenever possible, especially when the zbMath entry provides the DOI. Appropriate modification of old files is advised (upon editing them); e.g. the reference to Counterexamples in topology should be switched from the DOI to zb:0386.54001. Note that this requires change in the refs header and the links in text {{doi: ... }}.