There is no simple answer. It is just impossible to say “one day” or “one week”.
It depends on what wine it is.
Wine starts to change and then deteriorate from the moment it is exposed to air.
Wines which have higher levels of alcohol or sugar - because both are preservatives - keep longer. A bottle of Port or Sherry (both are fortified to a higher alcohol level) or a sweet dessert wine will last a lot longer than an ordinary table wine.
A red wine with high levels of tannin might actually improve when left open for a day because the air accelerates the aging process and helps to soften the tannins.
If you want to keep an ordinary table wine, put the cork back in the bottle as soon as possible (to reduce oxygen exposure) and place in your fridge door. Keeping it cool helps to prolong its life, just as the fridge preserves other foodstuffs you place in it. Leave it at room temperature and it doesn’t last long. By the way, this is why so many wines served by the glass in bars are so unappealing – they’ve been open too long.
I personally prefer to drink wines that have been opened on the day I drink it, but for this article I have been experimenting with keeping wines open and some big red wines, stored in the fridge have been drinkable after a week, although they didn’t have the freshness and fruit that I enjoyed when it was first opened.
If you want to finish the rest of the bottle more than a day later, try freezing the remainder. Put the cork back in the neck and place in freezer. When needed, thaw the wine slowly in the fridge. I might taste a little duller than when first opened, but it will taste a lot better than if you'd left it opened for several days. You might see what looks like dust or tiny bits of glass when thawed. These are harmless crystals of tartaric acid that have come out of solution in the cold.
If you intend regularly opening wines to finish later, see if you can find some smaller bottles -- those quarter sized screwcap bottles are ideal -- and pour the unused wine into the smaller bottle and close it, thus there is less oxygen in the bottle.
But why not look for wine in half or quarter bottles?
There are some devices that pump out the air -- but they are not very effective and many wine lovers think they ‘scrape’ out flavours. In my opinion they are a waste of money, and harm the wine. Some devices that pump nitrogen into the bottle to exclude oxygen, but these are expensive and I haven’t tried them myself.
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Peter F May is the author of Marilyn Merlot and the Naked Grape: Odd Wines from Around the World which features more than 100 wine labels and the stories behind them, and PINOTAGE: Behind the Legends of South Africa’s Own Wine which tells the story behind the Pinotage wine and grape, also available for the Kindle.