Really stupid question about mattress toppers

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After a stay in a hotel which had a mattress topper I have decided to get one for home, and very lovely it is too.
The thing is though that I had a mattress protector on the mattress before. Should I now put this over the topper?
The topper can be washed in the machine so I guess it isn't needed like it is for a mattress.

I found that my mattress protector wasn't deep enough to cover the mattress and the topper.
I think it would be a good idea to protect the topper if you can find a way though, without it affecting how nice it feels. I'm about to throw away a second mattress topper...I shall be experimenting too when I get another one!

The one I got can be split in two to make it washable in the machine.
I guess in a way this negates the need for a protector.
I just feel a bit odd only having a sheet between me and it I guess.

I have the mattress protector on the mattress and put the topper on top of that IYSWIM.