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I asked this on 온라인가나다, but 국립국어원 said this word is not on the standard dictionary and thus it cannot be answered: here.

닫린집합 is a term in Topology, and is a translation of 'clopen set'. Just like 'clopen' is a portmanteau of 'closed' and 'open', 닫린 is a portmanteau of 닫힌 and 열린.

I can't figure out what consonant assimilation happens to [ㄷㄹ]. The standard pronunciation rules (표준 발음법) doesn't present any example of such kind.

If the [붙임] of 제19항, "ㄹ following ㄱ or ㅂ is pronounced as [ㄴ]", is also applicable to ㄹ following ㄷ, 닫린집합 will assimilate to [닫닌지팝] and then to [단닌지팝].

Is this correct? My colleagues always pronounced it as [달린지팝].

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    Those kinds of sets are usually referred to as "열리고 닫힌 집합". I've never heard of someone call it 닫린 집합. And as far as I know, ㄷ is never followed by ㄹ in a single word, so there would be no context to compare this to. But if I were to pronounce it, despite its unnatural combination, I'd agree with your interpretation and pronounce it as [닫닌]→[단닌]. – Jihyung Kang Apr 17 at 8:42
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닫린집합 is not a standard word and no standard words have the ㄷㄹ part. Nonetheless, I believe that pronouncing 닫린 as 단닌 is reasonable because 곶리도, which is a standard word, is pronounced as 곤니도.

곶리도 → [리도] →(ㄹ-nasalization) [곧니도] →(ㄷ-nasalization) [곤니도]

린 →(ㄹ-nasalization) [닫닌] →(ㄷ-nasalization) [단닌]

As you can see in the comment section of this answer, some may argue that ㄷㄹ → [ㄴㄴ] has never been set down as a rule. They must not have read the part mentioning the reason 제19항 excludes ㄷ:

... 이 조항에서는 ‘ㅁ, ㅇ’과 ‘ㄱ, ㅂ’의 네 자음 뒤에서 이 현상이 일어난다고 규정하고 있다. 이처럼 4종류의 자음 뒤로 국한한 데에는 이유가 있다. ... 이 현상은 주로 한자어에서 일어나는데 한자 중에는 그 음이 ‘ㄷ’으로 끝나는 것이 없다. 이러한 사정들로 음절 종성에서 발음되는 7종류의 자음 중 ‘ㄴ, ㄷ, ㄹ’이 빠져 이 조항에서 언급한 ‘ㄱ, ㅁ, ㅂ, ㅇ’만이 남게 되는 것이다.

I deem that your colleagues have this logic:

닫린 →(ㄷ-nasalization**?**) [단린] →(liquidization) [달린]

or

달린 sounds natural because of the word 달리다.

According to 제18항, however, ㄷ gets nasalized before ㄴ or ㅁ; 제18항 does not say that ㄷ can get nasalized before ㄹ. In addition, 달리다 is irrelevant to 닫히다 and 열리다. For these reasons, I would say that [달린] is the incorrect pronunciation.

Edit: There was an interesting study (2019) on the pronunciation of the ㄷㄹ sequence. The abstract says the following:

/ㄴㄴ/ is considered to be the result of ‘ㄷㄹ’ sequence applied by ‘l-nasalization’ (forming the intermediate form 'ㄷㄴ') and ‘nasalization’. /ㄹㄹ/ is thought to be the result of the application of the ‘regressive liquidization’ of the /ㄷㄹ/ sequence. In the previous studies, the application environment of ‘regressive liquidization’ was limited to the ‘ㄴㄹ’ sequence only, but in this paper, the application environment of ‘regressive liquidization’ is claimed to has expanded to the ‘ㄷㄹ’ sequence.

So, people may have this simple logic:

닫린 →(regressive liquidization) [달린]

which is convincing, but this question remains: Is it correct to say that the pronunciation rules do not reflect the regressive liquidization? I would not accept such liquidization but admit that some people think of it as a common phenomenon these days.

Part of the conclusion says the following:

10·20대 화자가 발음한 ‘ㄷㄹ’ 연쇄에서 /ㄹㄹ/이 69%를 차지했고, /ㄴㄴ/은 29%에 불과했다. 그러나 이와 반대로 30·40대 화자는 ‘ㄷㄹ’ 연쇄 발음형에서 /ㄹㄹ/은 10%로 적게 나왔고, /ㄴㄴ/이 89%로 높게 실현되었다.

It says that most people in their 10s and 20s pronounced ㄷㄹ in 69% of the words as ㄹㄹ, whereas most people in their 30s and 40s pronounced ㄷㄹ in 89% of the words as ㄴㄴ.

The paper also shows that, as for 아웃렛 only, 95% of people in their 30s and 40s pronounced ㄷㄹ as ㄹㄹ. I believe that it is because 아울렛 has been used as if it were the correct spelling.

In 2018, National Institute of Korean Language answered a question on the pronunciation of the phrase 겉 라인 as the following:

‘겉’과 ‘라인’의 결합에 나타나는 ‘ㄷㄹ’의 표준발음이 무엇인지는 정해져 있지 않습니다. 뒷말이 ‘ㄹ’로 시작하는 것은 거의 ‘라인’과 같은 외래어에 한정되어 있고 외래어의 표준발음이 정해져 있지 않기 때문입니다. 한자어에서 ‘악력 [앙녁], 협력 [혐녁]’처럼 ‘ㄱ, ㅂ’ 뒤에서 ‘ㄹ’을 [ㄴ]으로 발음하는 것을 참고하면 ‘ㄷㄹ’도 [ㄴㄴ]으로 발음하는 것이 합리적이라고 볼 수 있겠습니다.

They said in the last sentence that it is reasonable to pronounce ㄷㄹ as ㄴㄴ, so they agree with you and me.

People, including us, have their own language habit and learn pronunciations from what others around them say rather than from what the rules say. I mean, the "standard" pronunciation is not the only pronunciation used in reality. Even if they pronounce 닫린 as 달린, you can still keep your way.

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The standard doesn't specify what happens when [ㄷ] is followed by [ㄹ]. So according to the standard, we can't answer the question of how 닫린 should be pronounced.

However, there are some words in a similar situation, namely 아웃렛. The National Institute of the Korean language Twitter answered the question of how 아웃렛 is pronounced, which basically said /ㄷㄹ/ can be pronounced either [ㄹㄹ] or [ㄴㄴ].

So according to this answer, both [달린지팝] and [단닌지팝] seems to be okay.

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