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NOTES WEEK 08 DAY 1
SCOT ADAMS
Let T be a set of sets. We recalled that T is a topology means:
xT yY “ T “ xT yfin X . Also, for any set X, T is a topology on X
Ť
means: ( T is a topology ) and ( T “ X ). Also, for any set X,
pX, T q is a topological space means: T is a topology on X.
Let pX, T q be a topological space and let A Ď X. Recall that, for
any x P X, by A is a nbd of x, we mean: DU P T s.t. x P U Ď A.
Recall: @set S of sets, @set A, we defined S|A :“ tS X A | S P Su.
Let pX, T q be a topological space and let A Ď X. Then the relative
topology on A inherited from pX, T q is T |A. (It is an unassigned
exercise to show that T |A is a topology on A.)
NOTE TO SELF: Next time I teach this, be sure to go over:
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Every
Every
Every
Every
Every
Every
Every
Every
subspace open extends to an ambient open.
subspace closed extends to an ambient closed.
subspace nbd extends to an ambient nbd.
subspace pnbd extends to an ambient pnbd.
ambient open restricts to a subspace open.
ambient closed restricts to a subspace closed.
ambient nbd restricts to a subspace nbd.
ambient pnbd restricts to a subspace pnbd.
Recall that:
BR :“ tpa, bq | a, b P R, a ă bu,
B˚ :“ BR Y tr´8, bq | b P Ru Y tpa, 8s | a P Ru,
TR :“ xBR yY ,
T˚ :“ xB˚ yY .
and
These are: the standard base on R, the standard base on R˚ , the
standard topology on R and the standard topology on R˚ .
REMARK 0.1. Let S be a set of sets and let A be a set. Then
we have pxSyY q|A “ xS|AyY and pxSyX q|A “ xS|AyX . Also, we have
pxSyfin Y q|A “ xS|Ayfin Y and pxSyfin X q|A “ xS|Ayfin X .
Date: October 25, 2016
Printout date: November 28, 2016.
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SCOT ADAMS
Proof. Omitted.
COROLLARY 0.2. Let pX, T q be a topological space and let B be a
base for T . Let A Ď X. Then B|A is a base for T |A.
Proof. Because B is a base for T , we conclude that xByY “ T . Then
xB|AyY “ pxByY q|A “ T |A. Thus B|A is a base for T |A, as desired. DEFINITION 0.3. For any A Ď R˚ , we define BA :“ B˚ |A and
TA :“ TA . These are called the standard base on A and standard
topology on A, respectively.
DEFINITION 0.4. We define N˚ :“ N Y t8u.
REMARK 0.5. Let A :“ t100, 101, 102, . . .uYt8u. Then A is a basic
open nbd of 8 in N˚ .
That is, using the standard base and topology on N˚ , we assert that
A is a nbd of 8 and that A is basic open. (Recall that “basic open”
means “an element of the base”.)
Proof. Since p99, 8s is a basic open subset of R˚ , it follows that p99, 8sX
N˚ is a basic open subset of N˚ . So, since A “ p99, 8s X N˚ , we see
that A is a basic open subset of N˚ . So, since 8 P A. It remains to
show that A is a neighborhood of 8 in N˚ .
Since 8 P A and since A is open in N˚ , we conclude that A is a
neighborhood of 8 in N˚ .
Note, also that t98u Y t100, 101, 102, . . .u Y t8u is an open nbd of 8
in N˚ , but is not basic open.
DEFINITION 0.6. Let pX, T q be a topological space and let a P X.
Let N be a set of nbds of a. Then N is a nbd base at a in pX, T q
means: @nbd S of a in pX, T q, DB P N s.t. B Ď S.
We often either abbreviate “in pX, T q” to “in X”, or simply omit
“in pX, T q”, if the context is sufficiently clear.
Some examples:
‚ Let a P R. Then tpa ´ ε, a ` εq | ε ą 0u is a nbd base at a both
in R and in R˚ . This nbd base is called the standard nbd
base at a.
‚ Let a P R. Then tpa ´ p1{nq, a ` p1{nqq | n P Nu is a countable
nbd base at a both in R and in R˚ . This nbd base is called the
standard countable nbd base at a.
NOTES WEEK 08 DAY 1
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‚ The set tpM, 8s | M P Ru is a nbd base at 8 in R˚ . This nbd
base is called the standard nbd base at 8.
‚ The set tpM, 8s | M P Nu is a countable nbd base at 8 in R˚ .
This nbd base is called the standard countable nbd base at
8.
‚ The set tr´8, ´M q | M P Ru is a nbd base at ´8 in R˚ . This
nbd base is called the standard nbd base at ´8.
‚ The set tr´8, ´M q | M P Nu is a countable nbd base at ´8
in R˚ . This nbd base is called the standard countable nbd
base at ´8.
‚ Let pX, dq, for every a P X, Then tBpa, rq | r ą 0u is a nbd base
at x in pX, Td q. This nbd base is called the standard nbd
base at x.
‚ Let pX, dq, for every a P X, Then tBpa, 1{nq | n P Nu is a countable nbd base at x in pX, Td q. This nbd base is called the
standard countable nbd base at x.
DEFINITION 0.7. For any set S, for any a, we define Saˆ :“ Sztau.
DEFINITION 0.8. Let pX, T q be a topological space, let P Ď X and
let a P X. Then P is a T -pnbd or, simply pnbd of a means: there
exists a neighborhood S of a in pX, T q such that P “ Saˆ .
In Definition 0.8, “pnbd” is short for “punctured neighborhood”.
For example, t100, 101, 102, . . .u is a pnbd of 8 in N˚ .
NOTE TO SELF: Next time, in Definition 0.9 below, say “for every
nbd V of b in pimrf sqYtbu, there exists a pnbd P of a in pdomrf sqYtau
such that f˚ pP q Ď V ”. Then follow up with an explanation (HW?) that
it’s equivalent to: “for every nbd V of b in Y , there exists a pnbd P
of a in X such that f˚ pP q Ď V ”.
DEFINITION 0.9. Let pX, T q and pY, Uq be topological spaces, let
f : X 99K Y and let a P X. Then, for all b P Y , the notation ( f Ñ b
near a ) means: for any nbd V of b in Y , there exists a pnbd U of a
in X such that f˚ pP q Ď V . Also, LIMSpf q :“ tb P Y | f Ñ b near au.
a
Also, lim f :“ ELTpLIMSpf qq.
a
a
When x is unbound, “f pxq Ñ b as x Ñ a” means “f Ñ b near a”,
with temporary binding on x. When t is unbound, “f ptq Ñ b as t Ñ a”
means “f Ñ b near a”, with temporary binding on t. Etc.
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SCOT ADAMS
When x is unbound, “lim f pxq” means “lim f ”, with temporary
xÑa
a
binding on x. When t is unbound, “lim f ptq” means “lim f ”, with
tÑa
a
temporary binding on t. Etc.
Example: Define a P RN by aj “ 1{j. Then a : N˚ 99K R˚ . We
played a game to show: ( a‚ Ñ 0 near 8 ). We played a game to show:
NOT ( a‚ Ñ 1 near 8 ). We argued that LIMSpa‚ q “ t0u. We con8
cluded that lim a‚ “ 0, or, equivalently, lim aj “ 0. More generally,
jÑ8
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we have:
FACT 0.10. Let a P RN . Then, for every z P R˚ , we have:
p a‚ Ñ z q
ô
p a‚ Ñ z near 8 q.
Also, LIMSpa‚ q “ LIMSpa‚ q. Also, lim a‚ “ lim a‚ .
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Proof. Omitted.
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