Erratum to “Topological complexity is a fibrewise L–S category” [Topology Appl. 157 (1) (2010) 10–21]

There is a problem with the proof of Theorem 1.13 of Iwase and Sakai (2010) [2] which states that for a fibrewise well-pointed space X over B, we have cat B B ( X ) = cat B ⁎ ( X ) and that for a locally finite simplicial complex B, we have TC ( B ) = TC M ( B ) . While we still conjecture that Theorem 1.13 is true, this problem means that, at present, no proof is given to exist. Alternatively, we show the difference between two invariants cat B ⁎ ( X ) and cat B B ( X ) is at most 1 and the conjecture is true for some cases. We give further corrections mainly in the proof of Theorem 1.12.