Is "closedness" a proper word?

Question

In one of my papers I had to prove a list of properties of a set, say, $S=\{a,b,c\}$. Among them we have a fact that $S$ is downward closed with respect to a binary relation $R$. I found it awkward to start proving the property by saying "Regarding downward closedness, the set $\{a,b,c\}$ is downward closed, since ... ." Is using the word "closedness" a good style or is there a better replacement? How would you reformulate the sentence?

Answer 1

Sure. The phrase “downward closedness” returns plenty of Google hits from math books. But your example feels clunky, because you’re essentially introducing the proof twice. Just eliminate the “regarding downward closedness” bit and you’re good to go! Some more alternatives:

• “$S$ is downward closed: …”
• “$S$ is downward closed. To show this, …”
• “To show that $S$ is downward closed, …”