Consider the following properties a space may satisfy:
(1) connected
(2) path connected
(3) locally connected
(4) locally path connected
(5) compact
(6) limit point compact
(7) locally compact Hausdorff
(8) Hausdorff
(9) regular
(10) completely regular
(11) normal
(12) first-countable
(13) second-countable
(14) Lindelof
(15) has a countable dense subset
(16) locally rnetrizable
(17) metrizable
For each of the following spaces, determine (if you can) which of these properties it satisfies. (Assume the Tychonoff theorem if you need it.)
(a)
(b)
(c)
(d) The ordered square
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(a)
The properties satisfied by
Solution:
The properties satisfied by
Given:
Consider the following properties a space may satisfy:
(1) connected
(2) path connected
(3) locally connected
(4) locally path connected
(5) compact
(6) limit point compact
(7) locally compact Hausdorff
(8) Hausdorff
(9) regular
(10) completely regular
(11) normal
(12) first-countable
(13) second-countable
(14) Lindelof
(15) has a countable dense subset
(16) locally rnetrizable
(17) metrizable
Approach:
Some basic results of topology for a space
Calculation:
Consider the space
Conclusion:
Hence, the properties satisfied by
(b)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Conclusion:
Hence, the properties satisfied by
(c)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Since,
Conclusion:
Hence, the properties satisfied by
(d)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Conclusion:
Hence, the properties satisfied by
(e)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Conclusion:
Hence, the properties satisfied by
(f)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Conclusion:
Hence, the properties satisfied by
(g)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Conclusion:
Hence, the properties satisfied by
(h)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Conclusion:
Therefore, the properties satisfied by
(i)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Consider
Conclusion:
Hence, the properties satisfied by
(j)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Conclusion:
The properties satisfied by
(k)
The properties satisfied by
Solution:
The properties satisfied by
Consider the space
Conclusion:
Hence, the properties satisfied by