# Generalized Uniform Boundedness Theorem

The following question is from the book "General Topology" written by John Kelly, Exercise 6.U in page 215.

Def: A meager set is a union of countably many nowhere dense sets.

Let $$X$$ be a real linear topological space which is not meager in itself and let $$K$$ be a closed convex subset of X such that $$K$$ = - $$K$$ and $$K$$ contains a line segment in each direction (i.e. for each $$x \in X$$ there is a positive real number $$t$$ such that for any $$s \in$$ [$$0, t$$], $$sx \in K$$. Show that $$K$$ is a neighborhood of $$0$$ (the identity element of $$X$$.

Used fact:

1): (From Exercise 6.P in page 211) A subset $$A \subseteq X$$ almost in $$X$$ (or satisfy the condition of Baire iff there is a meager set $$B$$ such that ($$A$$ \ $$B) \cup (B$$ \ $$A$$) (denoted as $$A \Delta B$$) is open.

2): (From Exercise 6.P.a) in page 211) A subset $$A$$ is almost open in $$X$$ iff there are meager sets $$B$$ and $$C$$ such that ($$A$$ \ $$B$$) $$\cup$$ $$C$$ is open.

3): (From Exercise 6.P.b) in page 211) For any subsets $$A$$ in a topological group ($$X, \tau$$), if $$A$$ contains a non-meager almost open subset, then $$AA^{-1}$$ is a neighborhood of the identity element.

Also 3) is known as the *Banach-Kuratowski-Pettis Theorem*.


According to the hint, I need to prove that $$K$$ is non-meager and almost open. The only way I came up with is to apply contradiction, assuming $$K$$ is meager. I mainly had difficulty using the condition "contains a line segment".

I added some random thoughts below ....

Assume $$K$$ is meager and hence $$K = \cup_{n \in \omega}U_n$$. Fix $$a \in X$$ and assume $$sx \in K$$ $$\forall s \in [0, t_x]$$. Since the interior of each $$U_n$$ is empty and there are only countably many of them, I believe this will imply one of $$U_n$$ will contain $$sx$$ $$\forall s \in [t_n, t_n^{'}]$$ where $$0 \le t_n^{'} < t_n < t$$.

To make it more clear, WLOG say $$K = \cup_{q \in \mathbb{Q_t}}U_q$$ where $$\mathbb{Q_t}$$ is the set of rationals in [$$0, t$$]. Fix $$U_k$$ and assume $$sx \in U_k$$ $$\forall s \in [(t^{'})_{x, k}, t_{x, k}]$$. Once I have the set $$\{t_{x, k}\}_{x \in X}$$, if its inf is $$0$$, then I can not find a neighborhood inside $$U_k$$

Any hints will be appreciated.

The "contains a line segment" condition implies $$\bigcup_{n\geq 1} nK= X.$$ Since $$X$$ is not meagre in itself, one of the sets $$nK$$ is non-meager, so $$K$$ is non-meager. Any closed set is almost open because it's Borel (or more directly, $$K$$ is the union of its interior and its boundary).
(I feel the use of 6.P is a bit convoluted. More directly: $$K$$ contains a neighborhood $$N$$ of a point $$x,$$ and contains $$-tx$$ for some $$t>0.$$ By convexity $$K$$ contains the neighborhood $$(tN-tx)/(t+1)$$ of $$0.$$)
• Thank you for your answer, Dap. I should have noticed that $nK$ cover the whole space and that is the main block I met. – Sanae Kochiya Jan 24 at 23:43