- #1
Bashyboy
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Homework Statement
If two different topologies on a given space ##X## are given by a basis, then we have the following criteria for determining which topology is finer:
Let ##\mathcal{B}## and ##\mathcal{B}'## be bases for the for topologies ##\tau## and ##\tau'##, respectively, on ##X##. Then the following are equivalent:
(1) ##\tau \subseteq \tau'##
(2) For each ##x \in X## and each basis element ##B \in \mathcal{B}##, there is a basis element ##B' \in \mathcal{B}'## such that ##x \in B' \subseteq B##.
Homework Equations
The Attempt at a Solution
I am wondering, do we have a similar criteria if the two topologies are given by a subbasis ##\mathcal{S}## and ##\mathcal{S}'##? My motivation for asking this question is that I am trying to show that ##\{Y \cap S ~|~ S \in S \}## forms a subbasis for the subspace topology on ##Y \subseteq X##, and I thought that it might help me in proving this.