Can someone confirm that a cross-London through ticket from, say, West Hampstead to the Brighton Line / south coast / etc would be valid via Kensington Olympia (and Clapham Junction etc) as well as via an Underground or Thameslink route across the centre. Thanks. I assume it's perfectly valid, but have learned [especially from this forum] not to take things for granted.
This ticket (West Hampstead Stations to Brighton, +Any Permitted) is valid from all three (Overground, Underground, Thameslink) West Hampstead stations to Brighton.
From West Hampstead (Overground) to Brighton, a valid route is via Kensington (Olympia), Clapham Junction and Gatwick Airport. This is the shortest route wholly by rail so no further check is needed.
From West Hampstead Thameslink to Brighton, a direct train is automatically valid, no further check is needed. If you want to change trains, the usual routes (such as via London Bridge and East Croydon) are listed in the Routeing Guide as valid routes.
From West Hampstead (Underground) to Brighton, it is valid by the provision of the cross-London transfer, as West Hampstead Thameslink to Brighton is valid via London as mentioned above, and West Hampstead (Underground) is a valid cross-London transfer tube station where you can start your tube transfer to places such as Victoria, London Bridge, Blackfriars, Balham, which is on a valid route for the National Rail journey between West Hampstead Thameslink and Brighton.
As a result, this is an extremely flexible, good-value ticket which can be used to travel between the majority of London Terminals with unlimited breaks of journey in between, including all the southern ones and also St Pancras, something which neither a ticket from London Terminals (not valid at St Pancras) nor London Thameslink (not valid at Victoria) can do, and also one tube journey for each of the outward and return between the appropriate terminals as well, with absolutely no operator restrictions (valid on the Gatwick Express as well!). At £17.70 before discount, you can, for example, travel from Brighton on the Gatwick Express to Victoria (cheaper than a normal Gatwick Express ticket from the airport!), then take the tube to Kings Cross to visit the British Library, have a walk through Camden Town and get back the train at Kentish Town, change at West Hampstead for the Overground to Shepherd's Bush for shopping before returning to Brighton, all with a return ticket on a weekend.