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- Thread starter lee.spi
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- #2

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1) First, you need to give an underlying set ##X##

2) Secondly, you need to give a set ##\mathcal{T}## which consist of subsets of ##X##. This set ##\mathcal{T}## is called the topology. It can be completely arbitrary, but it does need to satisfy the axioms of a topological space.

For example,

1) I give you the set ##X=\mathbb{R}##

2) I give the topology ##\mathcal{T} = \{\emptyset,\mathbb{R}\}##.

These two pieces of data specify a topological space, called an indiscrete space.

Other example:

1) I give you the set ##X=\{0,1\}##

2) I give the topology ##\mathcal{T} = \{\emptyset,\{0,1\},\{0\}\}##

These two pieces of data specify the Sierpinski topological space.

Obviously, not everything will be a topological space. For example

1) I give you ##X=\{0,1,2\}##

2) I give you ##\mathcal{T} = \{\emptyset, \{0,1,2\}, \{0,1\}, \{1,2\}\}##

These are again two pieces of data, but they do not specify a topological space because ##\mathcal{T}## does not satisfy the axioms. Indeed, ##\{0,1\}\cap\{1,2\} = \{1\}## is not in ##\mathcal{T}##.

- #3

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both “a distinguished collection of subsets”and “to be called the open sets” refer to the set T ?

- #4

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Yes, ##\mathcal{T}## is the "distinguished collection of subsets". Any element of ##\mathcal{T}## is by definition an open set.both “a distinguished collection of subsets”and “to be called the open sets” refer to the set T ?

- #5

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now i understand,thanksYes, ##\mathcal{T}## is the "distinguished collection of subsets". Any element of ##\mathcal{T}## is by definition an open set.

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