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When the author says that "women have no caste nor class," the meaning is that it is charm and poise, dignity, beauty and elegance (all superficial traits) that really put women on a pedestal. In other words, even the poorest girl with no class, without a famous last name, and without connections, could land herself a rich husband if she unites the physical traits that make the man look good.
This, however, is a focalized opinion based on how Mathilde views her role as a woman. We should greatly disagree with the way that the narrator makes the statement because such a statement is a conclusion from Mathilde's shallow mind. Moreover, the statement is greatly outdated since in today's modern, Western society, caste and class can take you far, but other variables can do just as much.
In today's world, a young woman like Mathilde would have found it much easier to make those "good marriage" dreams come true. It is much easier in today's society to make it to the top if you have the skills, the wits, the willingness, the opportunities, and the ability to make a talent, or a job skill, pay you back.
Look at all the people becoming rich developing apps, or creating products and services that others cannot create or are not willing to give. In other words, as of 2015, anyone with a plan can become rich, meet the right person, and perhaps even live a "dreamy" life.
Now, the necklace itself is a mirror image of Mathilde. It is flashy, it is pretty, it "looks" sophisticated enough, it can pass for a real thing--but it was none of those things. The necklace, flashy and pretty as it was, was a fake. It was made of paste and was not worth any money. No wonder Mathilde immediately connected with it out of all the pieces she saw: It reminded her of herself!
The necklace can also represent the injustice of the class system. Notice that, when Mathilde lost it, she had to figure out a way to pay it back. She endured anger, frustration, poverty, limitations and many, many sacrifices to pay it back--thousands of francs she had to pay for the replica of the necklace, which she went and got to put in place of the original.
She suffered intensely, feeling herself born for every delicacy and every luxury. She suffered from the poverty of her dwelling, from the worn walls, the abraded chairs, the ugliness of the stuffs. All these things, which another woman of her caste would not even have noticed, tortured her and made her indignant.
When she finds out that she paid so much for a lost necklace that was worthless in the first place, the reader has to wonder what exactly made Mathilde feel that she had a need to hide the truth from her friend and find another way to solve the problem.
The answer is found in her pride and shallow vanity, which are both symptoms that show that her society is unequal and not very tolerant. Being vain, egotistical, and too shy to admit that something is lost will just place her even lower (in her opinion) in her friend's eyes. Yet, had she not been vain, she could have openly told her friend about the loss. She could have offered ways to return the necklace, or even find something to replace with something else. Then, Mdme. Forrestier would have told her not to worry, that the necklace is a fake anyhow.
But Mathilde, as already stated, preferred the optics over the truth. She was always going to remain that shiny, perfect little jewel in the eyes of everyone. However, as already stated, she is a fake. She had to lie and make up an entire scenario for which she suffered a lot. None of Mathilde's pains would have even begun to occur, had she simply been true to herself, to her situation, and to her friend.