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The Wikipedia states where T-Money is valid: https://en.m.wikipedia.org/wiki/T-money

“Validity
Rapid transit systems in Seoul Capital Area, Busan, Daegu, Daejeon
Public transport bus systems in Seoul Capital Area (excluding Baengnyeong Island), Daejeon, Sejong, South Chungcheong, Daegu, Busan, Ulsan, Jeju etc.
Taxis that have a card reader (Seoul Capital Area, Daejeon, Busan, Daegu, Ulsan, Gwangju, Jeju etc.)”

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Also here: https://korea4expats.com/article-tmoney-seoul-transit-smart-card.html

“T-money cards are now accepted by transit systems in some areas outside Seoul. Therefore, you can use your T-money card on:
- all Seoul and Incheon buses & subway networks
- any Cheonan and Asan city buses with the T-money symbol in Chungcheongnam-do (province). 
Those without the T-money symbol do not accept T-money fare payment.
- all Wonju, Chuncheon and Gangneung city buses in Gangwon-do
- all Andong, Pohang, Gumi, Gwangju, and Mungyeong city buses in Gyeongsangbuk-do
- all Tongyeong and Geoje city buses in Gyeongsangnam-do
- all buses in Jeju.
- transportation systems in Busan”

It appears they are not valid in that area.