Relation To Other Topological Notions
- Separation
- Every product of T0 spaces is T0
- Every product of T1 spaces is T1
- Every product of Hausdorff spaces is Hausdorff
- Every product of regular spaces is regular
- Every product of Tychonoff spaces is Tychonoff
- A product of normal spaces need not be normal
- Compactness
- Every product of compact spaces is compact (Tychonoff's theorem)
- A product of locally compact spaces need not be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact is locally compact (This condition is sufficient and necessary).
- Connectedness
- Every product of connected (resp. path-connected) spaces is connected (resp. path-connected)
- Every product of hereditarily disconnected spaces is hereditarily disconnected.
Read more about this topic: Product Topology
Famous quotes containing the words relation to, relation and/or notions:
“Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.”
—Henry David Thoreau (1817–1862)
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“Hang ideas! They are tramps, vagabonds, knocking at the back- door of your mind, each taking a little of your substance, each carrying away some crumb of that belief in a few simple notions you must cling to if you want to live decently and would like to die easy!”
—Joseph Conrad (1857–1924)