sono uwasa ha matigai desita.

その means that and points to something close to the listener, already mentioned, or known from the context.

So その噂 means that rumor or the rumor we’re talking about.

A learner might compare it with:

• この噂 = this rumor
• その噂 = that rumor
• あの噂 = that rumor over there / that rumor more distant from both speaker and listener

In many real conversations, その often means the one you just mentioned.

は marks the topic of the sentence.

So その噂は means something like:

• As for that rumor, ...
• About that rumor, ...

It does not simply mark the subject in the same way English does. It sets up that rumor as the thing being talked about.

A very literal breakdown would be:

• その噂 = that rumor
• は = as for
• 間違いでした = was wrong / was mistaken / was false

間違い is a noun meaning mistake or error.

In this sentence, Japanese uses a noun + the copula:

• 間違いでした = was a mistake / was incorrect

This is very natural in Japanese. English often prefers was wrong or was false, but Japanese can express that idea with the noun 間違い.

So the sentence is not literally built like the rumor was wrong with an adjective. It is more like:

• That rumor was an error / was mistaken

Even though the natural English translation may sound more adjectival, the Japanese structure is nominal.

でした is the past polite form of the copula.

• です = is
• でした = was

So:

• その噂は間違いです。 = That rumor is wrong.
• その噂は間違いでした。 = That rumor was wrong.

In actual use, the past form can also suggest it turned out to be wrong after the facts became clear.

Usually yes, but in context it can feel a little broader than simple past tense in English.

For example, その噂は間違いでした can mean:

• the rumor was wrong
• the rumor turned out to be false
• that rumor was mistaken after all

So it often implies that someone believed or discussed the rumor before, and later learned it was incorrect.

They are similar, but not exactly the same.

• 間違いでした uses the noun 間違い
• 間違っていました uses the verb 間違う in a progressive/result form

Both can often be translated as was wrong, but the nuance is a little different.

その噂は間違いでした。

• sounds like That rumor was false / was a mistake

その噂は間違っていました。

• sounds more like That rumor was incorrect

The first is very common and natural when judging information as false. The second is possible too, but 間違いでした is often the more straightforward phrasing here.

Because その is a determiner, not a separate noun.

It directly modifies the noun that follows it, just like English that rumor.

So:

• その + 噂 = that rumor

You do not put a particle between them.

This works the same way with:

• この本 = this book
• その人 = that person
• あの映画 = that movie

噂 means rumor, gossip, or hearsay.

It often has the nuance of information spreading informally, not necessarily confirmed as true. Because of that, it can sometimes sound unreliable from the start.

It is not always strongly negative, but it often suggests:

• people are talking about something
• the information may not be verified
• it may be gossip rather than firm fact

So using 間違いでした with 噂 is very natural, because rumors are the kind of thing that can later turn out to be false.

You could change the particle, but the meaning and emphasis would shift.

For example:

• その噂は間違いでした。
  As for that rumor, it was false.
  This is the most neutral and natural version.

• その噂が間違いでした。
  This sounds more contrastive or identifying, like It was that rumor that was false.
  It is less neutral in this context.

So は is the most natural choice when simply making a statement about the rumor.

Yes. It is in the polite style because of でした.

Compare:

• その噂は間違いでした。 = polite
• その噂は間違いだった。 = plain

The polite version is suitable for normal conversation, reporting, and speaking respectfully. The plain version is more casual and would be used with friends, in informal writing, or inner thoughts.

A very literal breakdown is:

• その = that
• 噂 = rumor
• は = as for
• 間違い = mistake / error
• でした = was

So literally:

• As for that rumor, it was a mistake/error.

Natural English usually becomes:

• That rumor was wrong.
• That rumor was false.
• The rumor turned out to be false.

All of these can fit, depending on context.