What's wrong with this proof that the lower limit topology is finer than the K-topology?

general-topology

Solution 1:

$0 \in (-1,1)- K$, but not in your union $L$, so not in $L \cap [0,1)$ and not in $(L \cap [0,1)) \cup (-1,0)$ either.

Or, as the original argument said: $0$ is not an interior point of $\Bbb R- K$ in the lower-limit topology, while it is interior in the $K$-topology.

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