MiddlePath and Obaminator –
This is brought in Pirke Avos – “Hevei Makdim B’Shalom Kol Adam.” (I inquired with my Rav and he said this applies to whomever a person may pass in the street – men and women, Jewish or otherwise.)
Additionaly from Pirke Avos (1:15) – “Shammai omar: …m’kabayl et kol ha’adam b’sever panim yafot” (Shammai said: …receive everone with a thoughtful, pleasant countenance).
Lastly, it’s brought in the Gemara in Berachos 6b – “V’amar R’ Chelbo amar Rav Huna: Kol sh’yadua b’chavero sh’hu ragil l’yitain lo Shalom, yakdim lo Shalom…v’eem natan lo, v’lo h’chazeer, nikra gazlan (R’ Chelbo said Rav Huna said: All who know their friend will commonly greet him, he should precede him in greeting…and if (his friend) gives to him (a greeting) and he does not return (the greeting), he’s call a thief.”
Based on these three pearls from Chazal, we see the importance of greeting another whomever it may be.
I make it a point to be makdim b’Shalom each time I pass someone in the street by looking them in the eye, smiling, and recognizing that they too were created “b’tzelem Elokim.”