She was walking alone, always wearing the same béret, and always with the same white dog; no one knew who she was, and every one called her simply "the lady with the dog." (1.2)
Anna is very much separate from the rest of the Yalta crowd; she is isolated in the sense that no one knows who she is or anything about her.
One did not know what to do with oneself. (2.1)
Consider the way the environment drives action in this story…
I have been tormented by curiosity; I wanted something better. 'There must be a different sort of life,' I said to myself. I wanted to live! To live, to live! . . . I was fired by curiosity . . . you don't understand it, but, I swear to God, I could not control myself; something happened to me: I could not be restrained. (2.17)
It is ironic that Gurov is driven by the very same curiosity – yet Anna doesn't recognize this commonality and feels isolated from her lover anyway.
Gurov felt bored already, listening to her. (2.18)
Similarly, Gurov too fails to recognize that Anna is driven by the same "curiosity" that he himself felt earlier in Yalta.
He was moved, sad, and conscious of a slight remorse. This young woman whom he would never meet again had not been happy with him. (2.40)
Gurov is conscious of Anna's unhappiness, but he fails to recognize it as similar to his own dissatisfaction. The two lovers are further isolated by this lack of understanding between them.
Little by little he became absorbed in Moscow life, greedily read three newspapers a day, and declared he did not read the Moscow papers on principle! He already felt a longing to go to restaurants, clubs, dinner-parties, anniversary celebrations, and he felt flattered at entertaining distinguished lawyers and artists, and at playing cards with a professor at the doctors' club. (3.2)
Though not explicit here, Gurov's new loneliness is evident through his need for constant company.
Gurov went without haste to Old Gontcharny Street and found the house. Just opposite the house stretched a long grey fence adorned with nails.
"One would run away from a fence like that," thought Gurov, looking from the fence to the windows of the house and back again. (3.16-7)
This is the first time Gurov really makes an attempt to identify with Anna, to see the world from her perspective.
And he judged of others by himself, not believing in what he saw, and always believing that every man had his real, most interesting life under the cover of secrecy and under the cover of night. All personal life rested on secrecy, and possibly it was partly on that account that civilized man was so nervously anxious that personal privacy should be respected. (4.11)
This is a sad thought – how truly isolated each man would be if he had to hide that which was most valuable to him. It's no wonder Gurov feels as alone as he does.