I assume here that you mean inner cities in the US in the 1950s. The American inner city was in flux in the 1950s. US cities had witnessed rapid growth and development in the 1920s as a result of economic boom and large-scale immigration. This transformation was stalled by the Depression and the outbreak of World War II in the 1930s and 40s. The post-war US was witness to major growth, but much of it was leaving the cities behind.
Returning GIs weren't that interested in living in the urban centers, and the 1950s saw a massive explosion in the development of the suburbs. Everybody wanted their own home and little piece of the American dream. This left the inner cities to those who could not afford new homes or were excluded from the suburbs because of issues like racism. Many urban centers were characteristically ghettoized and heavily populated by African Americans and other new immigrants. Whites, in contrast, were fleeing to the suburbs, and the 1950s is the beginning of "white flight." The inner city thus became a place marked by crime, poverty, and urban decay. Many city centers, particularly in the Eastern US, witnessed significant decline.