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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.

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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 $0.$)

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  • $\begingroup$ Thank you for your answer, Dap. I should have noticed that $nK$ cover the whole space and that is the main block I met. $\endgroup$ – Sanae Kochiya Jan 24 at 23:43

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