What is meant by
    perfection in the hadeeth “no women attained perfection apart from four”?
Question

                Could you provide me with more
      information on the Hadith below. Jazka'Allah Khair 
      Narrated Abu Musa Al-Ash'ari (R.A.A): Allah's Messanger (S.A.W) said, "Many amongst
      men attained perfection but amongst women none attained the perfection except Maryam
      (Mary), the daughter of 'Imran, and Asiya, the wife of Fir'aun (Pharaoh). And the
      superiority of Aishah to other women is like the superiority of Tharid (i.e., and Arabic
      dish) to other meals." Sahih Al-Bukhari- Volume 5 (Book 62)
Praise be to Allah.Firstly: 
The scholars differed as to the meaning of the perfection of 
  women. Some said, it refers to Prophethood.
Ibn Hajar said in al-Fath:
 it is as if he said: No women attained 
  Prophethood except for So and so and So and so. (al-Fath, 
  6/447).
This view is mistaken!
The refutation of this idea is as follows:
Some reports also add  and Khadeejah 
  bint Khuwaylid and Faatimah bint Muhammad. (Narrated by at-Tabaraani). 
  We know for sure that Khadeejah and Faatimah were not Prophets, but they are 
  among the women who attained perfection. So what is meant by women who 
  attained perfection is the perfection of close friendship with Allaah 
  (wilaayah, the status of a wali), and not the perfection of Prophethood.
Al-Nawawi said:
Al-Qaadi said: this hadeeth is used as evidence by those who 
  say that a woman could be a Prophet and that Aasiyah and Maryam were Prophets!
But the majority say that they were not Prophets; rather, they 
  were Siddeeqahs [siddeeq: one who speaks or testifies to the truth] 
  and waliyyahs (close friends of Allaah) from among the awliyaa 
  of Allaah.
The word perfection is used to describe something 
  that is the best, the ultimate in its class.
What is meant here is that they achieved the highest degree 
  of virtue, righteousness and taqwaa.
And Allaah knows best. (Sharh Muslim, 
  15/198, 199).
Shaykh al-Islam [Ibn Taymiyah] said:
Al-Qaadi Abu Bakr, al-Qaadi Abu Yalaa, Abul-Maaali 
  and others mentioned that there was scholarly consensus that there could be 
  no female Prophet. The Quraan and the Sunnah do not indicate that, as 
  it says in the aayaat (interpretation of the meanings):
And We sent not before you (as Messengers) any but men 
  unto whom We revealed, from among the people of townships [Yoosuf 12:109]
The Messiah [Eesaa (Jesus)], son of Maryam (Mary), 
  was no more than a Messenger; many were the Messengers that passed away before 
  him. His mother [Maryam (Mary)] was a Siddeeqah [i.e. she believed in the Words 
  of Allaah and His Books [al-Maaidah 5:75]
Here it is mentioned that the highest degree his mother reached 
  was that of siddeeqah.
(Majmoo al-Fataawaa, 4/396).
Secondly: the hadeeth Faatimah is the 
  leader of the women of the people of Paradise except for Maryam bint Imraan. 
  (Narrated by Ahmad, 11347. Its isnaad was classed as hasan by al-Haafiz 
  Ibn Hajar in al-Fath, 7/111).
This hadeeth proves that Faatimah is better than Aasiyah. If 
  Aasiyah had been a Prophet, Faatimah would not have been better than her, because 
  Faatimah is not a Prophet. 
Thirdly:
Al-Karmaani said:
The word perfection does not necessarily imply 
  that she was a Prophet, because the word perfection is used to describe anything 
  that is perfect or is the best in its class. So it means that she reached the 
  highest degree in all the virtues that belong to women. (Al-Fath, 6/447). 
  This is the correct understanding of the perfection of women referred to in 
  the hadeeth.
Fourthly:
And the superiority of Aaishah to other women is 
  like the superiority of Thareed to other meals.
Ibn al-Qayyim said:
Thareed is composed of meat and bread. 
  Meat is the best of those foods that are eaten with bread, and bread is the 
  best of foodstuffs. If they are combined, there can be nothing better than that.
 Zaad al-Maaad, 4/271.
Al-Nawawi said:
The ulamaa said: it means that thareed is 
  better than broth, and thareed with meat is better than broth without 
  thareed, and the thareed that has no meat is better than broth. 
  What is meant by better is that it is more nutritious, more filling, easier 
  to digest, more enjoyable and easier to eat; a person may eat his fill of it 
  quickly, and so on. It is better than all kinds of broth and all kinds of food. 
  The superiority of Aaishah over other women is great, as great as 
  the superiority of thareed over other kinds of food. This does not clearly 
  state that she is better than Maryam and Aasiyah; it could be that what is meant 
  is that she is superior to the women of this ummah.
(Sharh Muslim, 15/199).
Ibn al-Qayyim said, discussing the relative merits of Aaishah 
  and Faatimah:
To say that one thing is better than another without discussing 
  it in detail is not right. If by better I mean the one who has more 
  reward from Allaah, this is something that cannot be known except through the 
  text or report (al-nass), because it has to do with the deeds of the 
  heart, not only the deeds of one's external faculties. How often have there 
  been two people, one of whom did more visible deeds, but the other was of a 
  higher status than him in Paradise.
If by better I mean the one who has more knowledge, 
  then undoubtedly Aaishah was more knowledgeable and brought more 
  benefit to the ummah, for she conveyed knowledge to the ummah such as 
  no one else conveyed and the leaders and rank-and-file of the ummah alike need 
  her. 
But if by better I mean of nobler birth and origin, 
  then undoubtedly Faatimah is better, for she is a part of the Prophet  
  (peace and blessings of Allaah be upon him). This is something unique, which 
  she shares only with her sisters.
If I mean leadership, then Faatimah is the leader of the women 
  of this ummah.
When discussing the question of superiority from all aspects 
  and approaches, this is fair and just. 
When most people speak about who is better or superior, they 
  do not discuss the matter in detail or take a balanced approach. This undermines 
  the truth, and if it is compounded by bias or whims, it becomes ignorant and 
  wrongful talk. 
(Badaai al-Fawaaid, 3/682, 683)
With regard to the virtues of Aaishah, they are 
  many (see question #7878). And Allaah 
  knows best.