Complementary topological properties

Question

‎Definition ‎1. A topological property ‎‎$‎P‎$‎ is expansive(contractive) if for any topology ‎$ ‎\tau‎$‎ with property ‎‎$‎P‎$‎ any finer (coarser) topology ‎$ ‎\tau‎^{*}‎‎ $‎ also has property ‎‎‎$ P $‎‎. ‎

Definition ‎2‎. If ‎‎$‎P$‎ is an expansive topological property and ‎‎$‎Q‎$‎ is a contractive topological property then ‎‎$‎P$‎ and ‎‎$‎Q‎$‎ are called complementary if a topology is minimal ‎‎$‎P$‎ if and only if it is maximal ‎‎$‎Q‎$‎.‎‎

‎ Definition ‎3‎. A topological space is called $KC$ space if every compact subset is closed.

Theorem: X‎ ‎is ‎‎maximal ‎compact ‎‎‎‎$ ‎\Longleftrightarrow‎ $‎‎‎ ‎‎‎‎X‎ ‎is ‎‎KC‎‎-‎minimal.

Therefore, $KC$ and compactness are complementary.

1: What property can be placed instead of compactness or $KC$ to complement?

2: Do you give me some examples of topological properties to complement each other?

Answer 1

Let $P$ be an expansive property and let $Q$ be a contractive property. There are different types of complementarity relating the following induced poperties: (1) minimal $P$, (2) maximal $Q$, (3) $P$ and $Q$.

The definition of complementary properties you mention (iirc defined by Larson) demands the equivalence of minimal $P$ and maximal $Q$, i.e. (1) $\iff$ (2). Of course a topology that satifies the equivalent properties is also both $P$ and $Q$, but I do not see a reason for the converse. So there may be complementary properties $P$ and $Q$ such that being simultaneously $P$ and $Q$ does not imply being minimal $P$ (and maximal $Q$).

It seems that often you start with properties $P$ and $Q$ such that being simultaneously $P$ and $Q$ implies both minimal $P$ and maximal $Q$, i.e. (3) $\implies$ (1) and (2). Equivalently, there is no $P$ topology strictly coarser than a $Q$ topology. Let us call such properties opposing. It may additionaly happen that minimal $P$ already implies $Q$ (i.e. (2) $\implies$ (3)) and/or that maximal $Q$ already implies $P$ (i.e. (1) $\implies$ (3)). In the best case, all conditions (1), (2), (3) are equivalent, and in particular the properties $P$ and $Q$ are complementary.

Some examples of opposing properties:

• $T_2$ and compact – the most classical example. But neither of the extra conditions is true. There is even a maximal compact topology strictly coarser than minimal Hausdorff topology.
• $KC$ and compact – this fixes the previous situation. Showing that maximal compact is $KC$ is easy. Showing that minimal $KC$ is compact is much harder, and it is quite recent result.
• door and connected. A topological space is door if every its subset is open or closed. These spaces are fully classified. Moreover, every minimal door space is connected. On the other hand, not every maximal connected space is door.
• door and hyperconnected. A topology is hyperconnected if every two nonempty open sets meet, i.e. the topology (without the empty set) is a base of a filter. Door hyperconnected topologies are just ultrafilters, so we have the equivalence with maximal hyperconnected spaces. But not every minimal door space is hyperconnected. Also the excluded point topologies are minimal door.
• $T_1$ and having only finite proper closed sets. This one is easy since on every set there is the coarsest $T_1$ topology – consisting of all cofinite sets (and the empty set). We have all equivalences here.
• $T_D$ and nested. $T_D$ is a separation axiom between $T_0$ and $T_1$ defined by the condition that every singleton is open in its closure. A topological space is nested if every two open sets are comparable by inclusion. It is equivalent to being hereditarily connected. Here we have the equivalence of all three conditions (1), (2), (3). And spaces satisfying these are Alexandrov discrete with the specialization preorder being a linear order.
• $T_0$ and strongly loosely nested. We have all equivalences here. Strongly loosely nested is a strengthening of nested.