Question

Is all finite topological space connected?

Answer 1

ʜɪɪ ᴍᴀᴛᴇ ʜᴇʀᴇ's ᴜʀ ᴀɴsᴡᴇʀ....

ɪɴ ᴀɴʏ ᴛᴏᴘᴏʟᴏɢɪᴄᴀʟ sᴘᴀᴄᴇ, ɪғ x ≤ ʏ ᴛʜᴇɴ ᴛʜᴇʀᴇ ɪs ᴀ ᴘᴀᴛʜ ғʀᴏᴍ x ᴛᴏ ʏ. ... ᴛʜᴇʀᴇғᴏʀᴇ, ᴀ ғɪɴɪᴛᴇ sᴘᴀᴄᴇ ɪs ᴄᴏɴɴᴇᴄᴛᴇᴅ ɪғ ᴀɴᴅ ᴏɴʟʏ ɪғ ɪᴛ ɪs ᴘᴀᴛʜ-ᴄᴏɴɴᴇᴄᴛᴇᴅ. ᴛʜᴇ ᴄᴏɴɴᴇᴄᴛᴇᴅ ᴄᴏᴍᴘᴏɴᴇɴᴛs ᴀʀᴇ ᴘʀᴇᴄɪsᴇʟʏ ᴛʜᴇ ᴘᴀᴛʜ ᴄᴏᴍᴘᴏɴᴇɴᴛs. ᴇᴀᴄʜ sᴜᴄʜ ᴄᴏᴍᴘᴏɴᴇɴᴛ ɪs ʙᴏᴛʜ ᴄʟᴏsᴇᴅ ᴀɴᴅ ᴏᴘᴇɴ ɪɴ x....

ʜᴏᴘᴇ ɪᴛ ʜᴇʟᴘs ᴜ....!!....

ᴘʟᴢᴢ ᴍᴀʀᴋ ᴀs ʙʀᴀɪɴʟɪᴇsᴛ...!!......

Answer 2

$\huge\underline{\red{⛥❃꧁αղsw៩Ʀ꧂⛥❃}}$

ɪɴ ᴀɴʏ ᴛᴏᴘᴏʟᴏɢɪᴄᴀʟ sᴘᴀᴄᴇ, ɪғ x ≤ ʏ ᴛʜᴇɴ ᴛʜᴇʀᴇ ɪs ᴀ ᴘᴀᴛʜ ғʀᴏᴍ x ᴛᴏ ʏ. ... ᴛʜᴇʀᴇғᴏʀᴇ, ᴀ ғɪɴɪᴛᴇ sᴘᴀᴄᴇ ɪs ᴄᴏɴɴᴇᴄᴛᴇᴅ ɪғ ᴀɴᴅ ᴏɴʟʏ ɪғ ɪᴛ ɪs ᴘᴀᴛʜ-ᴄᴏɴɴᴇᴄᴛᴇᴅ. ᴛʜᴇ ᴄᴏɴɴᴇᴄᴛᴇᴅ ᴄᴏᴍᴘᴏɴᴇɴᴛs ᴀʀᴇ ᴘʀᴇᴄɪsᴇʟʏ ᴛʜᴇ ᴘᴀᴛʜ ᴄᴏᴍᴘᴏɴᴇɴᴛs. ᴇᴀᴄʜ sᴜᴄʜ ᴄᴏᴍᴘᴏɴᴇɴᴛ ɪs ʙᴏᴛʜ ᴄʟᴏsᴇᴅ ᴀɴᴅ ᴏᴘᴇɴ ɪɴ x....