- #1
kent davidge
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I'm taking some study notes on general math, one of the topics being topological spaces.
I've had a look at the definition of what is understandable by countable in Mathematics and I arrived at the following definition for what is a second countable space
- Consider a topological space ##(X, \tau)##
- Suppose that ##\tau = \cup_{i \ \in \ G \ \subseteq \ \mathbb{N}} \{\tau_i \}## and that for any ##i##, ##\{\tau_i \} = \cup_{i \ \in \ F \ \subset \ \mathbb{N}} \{\tau_j \}##. Then ##\tau## has a countable basis and ##(X, \tau)## is second-countable in the topology ##\tau##. (Note that ##G## may be the whole of ##\mathbb{N}## whereas ##F## has to be a subset of ##\mathbb{N}##.)
Is this definition ok?
I've had a look at the definition of what is understandable by countable in Mathematics and I arrived at the following definition for what is a second countable space
- Consider a topological space ##(X, \tau)##
- Suppose that ##\tau = \cup_{i \ \in \ G \ \subseteq \ \mathbb{N}} \{\tau_i \}## and that for any ##i##, ##\{\tau_i \} = \cup_{i \ \in \ F \ \subset \ \mathbb{N}} \{\tau_j \}##. Then ##\tau## has a countable basis and ##(X, \tau)## is second-countable in the topology ##\tau##. (Note that ##G## may be the whole of ##\mathbb{N}## whereas ##F## has to be a subset of ##\mathbb{N}##.)
Is this definition ok?
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