Jewish communities don't actually require a synagogue as community prayer can take place anywhere. In some communities, the 'synagogue' is in someone's home. Synagogues are established as a place for community worship, gatherings, and study.
Synagogues traditionally have played a central role in Jewish life, just as churches and mosques have played an important role in Christian and Muslim life. They are not only centers of prayer, but of learning and study, of community and connection, including childhood and adult education as well as social activities such as arts, crafts, lectures, concerts and other community events. Because synagogues are buildings built on land and are generally not funded by any central authority, they cost money, to build, to maintain, to staff. So membership in and dues paid to a congregation helps support the facility and its work.
Synagogues also join together in relatively loose confederations; one for example for Reform Synagogues is the UAHC - Union of American Hebrew Congregations. These organizations are able to then help manage and promote content and consistency and provide additional services such as rabbinical schools, religious school curricula, summer camps, publication needs (prayer books, school books, etc.), other kinds of coordinated efforts including support for Israel, and a greater presence for the overall community politically than a single congregation could make.
On the other hand, technically wherever one can find ten Jewish adults, a prayer service is considered "official". And this does not prevent Jews from individual prayer. But the historical role of the synagogue is to emulate the great Temple of Solomon in Jerusalem as the Jewish people have moved into diaspora (spreading out over many lands outside the historic home of the Jewish people in Israel).
Synagogues are places where Jews meet for worship and study.
It depends entirely on the community. Smaller synagogues have communities of a few dozen families, larger synagogues can have up to one thousand families.
Jews worship in synagogues. there are synagogues wherever Jews live throughout the world.
Jews worship God whether or not synagogues are available.
No. Mosques are used by Muslims as their place of worship, not by Jews. Jews pray in synagogues.See also:More about Jewish prayer and synagogues
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"Judaism" is not something that is itself venerated. Jews worship God and do so by the rites and rituals prescribed by Judaism. Jews pray at Synagogues.
Yes, ALL Orthodox synagogues do, but in the other synagogues, some non-Orthodox Jews will wear their own.
P. Riera Vidal has written: 'The Jews in Toledo and their synagogues' -- subject(s): Synagogues, Jews, History