In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.
In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, "negligible". Some examples are finite sets, smooth curves in the plane, and proper affine subspaces in a Euclidean space. A topological space is a Baire space if it is "large", meaning that it is not a countable union of negligible subsets. For example, the three dimensional Euclidean space is not a countable union of its affine planes.
The precise definition of a Baire space has undergone slight changes throughout history, mostly due to prevailing needs and viewpoints. First, we give the usual modern definition, and then we give a historical definition which is closer to the definition originally given by Baire.
We don't need a party
just a place to meet
My mind is still racing while the city sleeps
But I don't complain
Cause I know we're okay
Race against the clock
Running on no sleep
When there's no time to rest
and there's no need to dream
Cause when your with me
I feel like I'm dreaming
I hang out and hold on
(and count the days til I come home)
I can't brave this alone
Quarantine My Heart Baby
In someone elses arms, I feel unhealthy
How can we survive and finally live our dreams
In a world filled up with players on opposing teams?
Darling how can I go on?
Sing me one more song
Not even an hour on the interstate
Such an unforgetting journey
An eternal wait
And I hit the cruise control
There's a swelling in my soul
It's so good
I speed up and get home
To find you again
Quarantine My heart again
In someone elses arms
I feel unhealthy
(So never let me go)
If I can't feel your heart beating
I might as well just write off love completely
Clogging up my mind, gotta get out
Open my mouth and let it out
I've tried and trust me,
It's not the same with anyone but you
I swear to God it's true
Quarantine my heart baby
In someone elses arms
I feel unhealthy
(so never let me go)
If I can't hear your heart beating
I might as well just write off love completely
With so little sleep
At least you'd think I'd find some peace in my dreams
In my dreams
But my mind still winds up on the same thing
The same scene
The same themes
Cause it's all stuck in my subconscious
Built up from every day
So I'm stuck with these nightmares
Where you're gone and so far away
And when I wake up
I realize that everything's still wrong
I'm still here and you're still gone
It's not fair
Cause either way I spin it
Separation seems so wrong
These breaks are far too long for me
Hours and hours
I'm stuck inside this place and this town
And you're gone
Far away, you're fighting for your life all alone
I want to wake up and go home
Cause it's all stuck in my subconscious
Built up from every day
So I'm stuck with these nightmares
Where you're gone and far away
Oh, this tortures me so much that
I get sick and I throw up
In my dream and here on my bed
It's messed up how it's all in my head
Yet it's affecting me oh so bad
I guess this distance just makes me sick
Cause when I wake up
It's 4 AM and I'm still all alone
Your message on my phone
Don't tell me that sleeping through the night
Is never this hard when you're home
Cause I already know
Wake me
Pick me up, I'm ready
And take me down, downtown
Down the underground
Shaking, can't get steady
Baby, calm me down
Calm me down
(This addiction knocks upon my door)
This addiction is knocking upon my door
I crawled back to you, sweating through my clothes
Pick me up, I'm ready
And take me down, downtown
Down the underground
Shaking, can't get steady
Baby, calm me down
Calm me down
Sing along with me
Electricity is running through our feet
Every time we meet
Look out, look out, look out, look out
When these two get together
Tonight, tonight, look out tonight
This night goes on forever
I made a promise to myself that I'd clean up
But every time I get a taste, I can't get enough
And now I'm on my knees
(And I smell electricity)
And I ain't even saying please
(Give me some ecstasy)
Sing along with me
Electricity is running through our feet
Every time we meet
Look out, look out, look out, look out
When these two get together
Tonight, tonight, look out tonight
This night goes on forever [x2]
In your position with this hometown competition
You're away
At least that's what the magazines all say
You drive the streets
Where he let his love deplete
We're one and the same
Two suburbanites looking for someone to blame
Maybe right now you look bored enough to die
Tomorrow you'll be getting on a plane
And I don't know why
You won't be smiling on the flight home
When you know that
He left you, left you alone in limbo
So disconnect the phone
And let your sorrow stretch from here to LA
Cause you know that I'm still here
And I know it's been a hard year
And it's clear that your life is changing still
So let your smile stretch from here to LA
I know you're trying
And at night you stay up, crying
You're awake
You shut your ears and realize your mistake
The West Coast sunset
Doesn't seem like so much fun yet
Does it now?
So you climb the stairs and search for your way out
And I've never hurt you
Like the boy you called your savior
Darling, do me just one favor
Look what a mess he's made you
Chicago's just a phone call away
I'll be up all night
There doesn't seem to be a solution
That makes this right
You won't be smiling on the flight home
When you know that
He left you, left you alone in limbo
So disconnect the phone
And let your sorrow stretch from here to LA
Cause you know that I'm still here
And I know it's been a hard year
And it's clear that your life is changing still
So let your smile stretch from here to LA
You let him in
Believe your skin, oh
Baby, I know you'll survive
You may be down
But you're breathing and you're still alive
You let him in
Believe your skin, oh
Baby, you're still breathing
Baby, you're still breathing
You won't be smiling on the flight home
When you know that
He left you, left you alone in limbo
So disconnect the phone
And let your sorrow stretch from here to LA
Cause you know that I'm still here
And I know it's been a hard year
And it's clear that your life is changing still
So how's your affair
With the millionaire with silver hair?
Does he care?
Your love affair, I am aware
Is void of love indeed
The greatest weight of all
Towers over your shoulders
While the longest wait of mine
Keeps playing over and over
These are simple times
And now I miss your simple pleasures
A substance state of mind
What I had left to treasure
I know it's been a while
But I swear you wouldn't be so lonely
If only you'd let me in
So look at you now
The spark in your eyes is gone, somehow
I wish you would hang up and turn around
And let me see those eyes
They'd be the same brown eyes
It would make my life to see you turn
I'd finally make my case
You'd--you'd run away and come with me
To destroy the human race
The greatest weight of all
Towers over your shoulders
While the longest wait of mine
Keeps playing over and over
These are simple times
And now I miss your simple pleasures
A substance state of mind
What I had left to treasure
I know it's been a while
But I swear you wouldn't be so lonely
If only you'd let me in
So look at you now
The spark in your eyes is gone, somehow
I wish you would hang up and turn around
And let me see those eyes
They'd be the same brown eyes
You're not here
Every night is a nightmare
So look at you now
The spark in your eyes is gone, somehow
I wish you would hang up and turn around
And let me see those eyes
They'd be the same brown eyes
I met a girl from the northwest suburbs, oh oh
She had a mouth like you wouldn't believe
She said, "Hey kid, I think I wanna get ccccrazy
So let's go
Let go of everything you want
And let me show you what you need"
Let's mix some drinks
And mix it up for a while
Think you're ready
To see the fires we ignite
When we go out at night?
Got in my car
Headed eastbound on Kennedy
Staring at the skyline on my way downtown
She hit the floor before I knew what h-hit me
Eighty-sixed after six drinks mixed
I said “ (c’mon) follow me”
Let's mix some drinks
And mix it up for a while
Think you're ready
To see the fires we ignite
When we go out at night?
I've got some things that I should clear up
But I don't think you're ready
So pass the bottle
Let's get up and go all out, all night
It's a disco, baby
(It's a disco, baby)
Don't you know we're supposed to dance like that?
Don't you know we're supposed to dance like that?
It's a disco, baby
(It's a disco, baby)
Don't you know we're supposed to dance like that?
Girl, you know you want to, want to, want to
Let's mix some drinks
And mix it up for a while
Think you're ready?
Oh yeah, we're fucking ready
Hell yeah, we're fucking ready
Fuck yeah, we go out all out, all night
Let's mix some drinks and mix it up
Oh yeah, we're fucking ready
Hell yeah, we're fucking ready
The cigarettes and phone calls don't keep me warm
But they're my only option
Short of driving home
To try and calm my nerves of this unrest
Cause nothing's worse than a vacant town, yeah
Still looking for the answers that I haven't found
With each new fact I'm learning
My mind feels like it's turning away from you
(So cut the ropes and let's go)
Train, take me away
Anywhere but here
I've had it
I just can't stand it
I'm seeing how quiet I can say your name
I whisper to no one just to keep me sane
An echo in the dark I wish was you
If I only knew how to form a sound
To convince myself you'll come around
These memories are fading
But I'm getting sick of waiting to hear from you
Train, take me away
Anywhere but here
I've had it
I just can't stand it
Please, it makes me insane
Everyday I just can't stand it
I just can't stand it
Train, take me away
Anywhere but here
I've had it
I just can't stand it
Please, it makes me insane
Everyday I just can't stand it
I just can't stand it
Please, train, take me away
(Please, train)
I gotta get outta this town
(Please, train)
If you could see me here this way
Looking backwards from today
Would you do it all again?
If I could roll it back to you
Just like lovers always do
I'm stoned in love, but not with you
Good luck
Cause we're not feeling stoned in love
It's how I am feeling now
I feel this love is made of stone
We're not feeling stoned in love [x2]
If we could wash the past away
Would we go our separate ways
Or do it all again?
As if I care
Is this the way it's meant to be
Stoned in love, but not with me?
You're stoned in love, but not with me
Good luck
Cause we're not feeling stoned in love
It's how I am feeling now
I feel this love is made of stone
We're not feeling stoned in love [x2]
Stoned in love
I'm stoned in love, but not with you
Good luck
Because we're not feeling stoned in love
It is how I am feeling now
I feel the love is made of stone
We're not feeling stoned in love
Good luck
Stoned in love [x6]
Stoned in love, but not with you
My feet are shuffling
The clouds roll in, the wind at my neck
But every day's a thunderstorm
I'm always walking back these days
I live to pass time
It makes sense in my mind
But that's not a reason
To keep this weak heart beating
And I know that this place
And these walls and this race
Will melt down in one breath
And leave it all behind me
My feet are shuffling
The clouds roll in, the wind at my neck
But every day's a thunderstorm
I'm always walking back these days
My knees are quivering
The cliff's edge so inviting, yeah
The lightning hits my chest
But the shock is so routine for me these days
You stand up and I bow
Right on, right now
I'm always hoping for some enjoyment from disappointment
I can feel the misery from some place deep inside of me
My feet are shuffling
The clouds roll in, the wind at my neck
But every day's a thunderstorm
I'm always walking back these days
My knees are quivering
The cliff's edge calls me
The lightning hits my chest
But the shock is so routine for me these days
You make my bed, I'll make your day
I'll write down every word you say
And stuff it in a record sleeve
And make sure no one ever, ever finds it
Can you save me now?
Someone, somehow
I want a box around myself
So they can put me on the shelf
To see the way I live
But the holes are blocked
By everybody else inside
As far as I can tell
They're all living well
Living better, still
Oh, that's the way it is
Climbing up the corporate ladder
Trying to escape through the roof
Can you explain lives led in vain?
It seems like everybody's looking for their way out
It's not a problem, it's hell
We only do what people tell
But can you tell me, anyone
The consequences of setting someone free?
Can you explain lives led in vain?
It seems like everybody's looking for their way out
Have you ever felt the pinch
When you life's confined?
Well, get your ass in line
It's hard not to care what they say
I'd like to think we'll break away
I don't believe that growing up
Means cashing out and giving up
Can you explain lives led in vain?
It seems like everybody's looking for their way out
This box contains just one thing
The fact that I will probably never find a way
Can't you help me fade out?
I found it difficult
To be a storytelling failure
For all those years
Living was a challenge
That caught me so off-balance
And passed me by
Repeatedly, I stared my future in the face
She walked away
Her silhouette confirming my worst predictions
She walks away
Hope is just four disconnected letters
Combined and put together
By a disconnected man who missed the point
My life was just a promise that I uttered, then I broke it
And admitted as we all went up in flames
So I made a resolution that I'd pay back planet earth
And I would send out this faded portrait
Maybe we would all live on
We'd soar on
So I strapped it to a rocket and I sent it into space
Praying someday you would find it
Maybe we would all live on
We'd soar on
In all my years existing
As a storytelling failure
For all those years
I waited for a miracle
On a planet based on push and pull
And nothing changed
And one day, someone spoke inside of me
"Young man, rise and make your mark"
So today I will be free
With no one there to see
With no one there to see
Hope is just four disconnected letters
Combined and put together
By a disconnected man who missed the point
My life was just a promise that I uttered, then I broke it
And admitted as we all went up in flames
(So I made a resolution...)
So I made a resolution that I'd pay back planet earth
And I would send out this faded portrait
Maybe we would all live on
We'd soar on
So I strapped it to a rocket and I sent it into space
Praying someday you would find it
Maybe we would all live on
We'd soar on
So I strapped it to a rocket and I sent it into space
Praying someday you would find it
Maybe we would all live on
Dude What Happened Last night...?
Lightening Strikes Me Down
I'm On my knees
I can't write it out
But I'll show you what I mean
and the words they will drip down your cheeks
and im able to read them out and they spell out
"Lets Just Be Friends"
Four simple words that we keep ignoring
Again and Again as we...
(As is we never meant it)
Baby, lets leave them all for dead,
All those thoughs inside your head
Can be realitiy so let yourself believe
My life is a cross
Of a fire and a stampede
I'll burn inside, you still run after me
Yeah, we'll find out, how it feels to burn alive
You and me girl, let's set ourselves on fire
(as if we never meant it)
Baby, Lets leave them all for dead
All those thoughts inside your head
They can be reality, just let yourself believe
Oh yeah, we don't care who finds out,
Tonight is all that matters right now
Let's go, once more, down to the dance floor
Make me sell my soul for more
(even when we fall alseep together)
(even when we fall lseep together)
(as if we never even meant it)
Baby let's leave them all for dead
All those thoughts inside you head
Can be reality, just let yourself believe,
I feel your body radiate, in the lighteining we create
So let yourself believe
Oh yeah, we don't care who finds out, tonight is all that matters right now
Let's go once more, down to the dance floor
Look at where you've got yourself now
You think the world is trying to bring you down
You think that moving on
Means getting passed around
But every move you make
Just takes you further down
These are simple times
And now I miss your simple pleasures
A substance state of mind
What I had left to treasure
I know it's been a while
But I swear you wouldn't be so lonely
If only you'd let me in
So look at you now
The spark in your eyes is gone, somehow
I wish you would hang up and turn around
And let me see those eyes
They'd be the same brown eyes
I can't help thinking so much about
How such a girl could dig a hole so deep
You can't find a way to get out
I'm not a helping hand
I'm a rope to pull you through
Just give me, give one more chance
To turn you back to you
Take this feeble mind
And mold it into something better
And help me realize that I won't live forever
I know it's been a while
But I swear you wouldn't be so lonely
If only you'd let me in
So look at you now
The spark in your eyes is gone, somehow
I wish you would hang up and turn around
And let me see those eyes
They'd be the same brown eyes
You're not here
Every night is a nightmare
So look at me now
The spark in my eyes is gone, somehow
I wish I could stand up and turn around
And let you see my eyes
(Oh my God, I can see into the future!)
Oh, I've been counting on this countdown, countdown
Only minutes till sundown, sundown
Do I trust the team in front of me?
Oh, can, can I stop and catch my breath, my breath?
Will this mission bring my death, my death?
Oh, can I run and catch my enemy?
Oh, I feel something waiting for me
Every time I start to breathe
Check my pulse
My vision's dimming
Catch my soul before it leaves
Lethal weapons, we're flaunting lethal weapons
(But I can't seem to hit my mark)
In the future, we're dancing in the future
(Why can't he seem to hit his mark?)
It's been so long since he's had a break, yeah
He's in the cave and he can't escape, yeah
The events I see will cause a tragedy
Oh, I, I needed you to trust me, trust me
But you think that I'm crazy, crazy
But there's just some things that you can't know
And I, I'm stopping crime with a new face, new face
Got the secrets in the briefcase, briefcase
We got a mystery
It can't get out
When I opened my eyes
I knew it was a dream
The future's not what it seems
Lethal weapons, we're flaunting lethal weapons
(But I can't seem to hit my mark)
In the future, we're dancing in the future
(Why can't he seem to hit his mark?)
It's been so long since he's had a break, yeah
He's in the cave and he can't escape, yeah
I'm just a man who can't be stopped
Baby, that's me
People say that there is no way out
(What's in the briefcase?)
Hold your breath
Keep the words in your mouth
(What's in the briefcase?)
Don't let the secrets out
They'll set you free
Lethal weapons, we're flaunting lethal weapons
(But I can't seem to hit my mark)
In the future, we're dancing in the future
(Why can't he seem to hit his mark?) [x2]
One, two, three
Are you feeling me?
It's getting late and it's hard to see
I've got some problems
But not enough to stop me
Cause I can't stop, I can't stop
I won't stop, won't stop
Can't stop until I know for sure
This night ends with me and you
(Be, be aggressive, be, be aggressive)
Be, be aggressive
B-e, be aggressive
You never know just who you're up against, so
B-e, b-e aggressive
B-e, b-e aggressive
Yeah, are you reading me?
Is it obvious what I'm trying to say?
Cause I want you bad
And you know I got it
Oh, I can't stop, I can't stop
I won't stop, won't stop
Can't stop until I know for sure
This night ends with me and you
Be, be aggressive
B-e, be aggressive
You never know just who you're up against, so
B-e, b-e aggressive
B-e, b-e aggressive
Cause you know you can't stop the thunder
Oh, I can feel the tension rising
But you know we've got it good
Who would have guessed
That tonight we'd take over the world?
B-e, be aggressive
B-e, be aggressive
You never know just who you're up against, so
B-e, b-e aggressive
B-e, b-e aggressive
(Chorus)
Alam kong nagkamali ako
Nagkamali ako sayo
Di ko naman sinasadya sanay mapatawad mo at sanay bigyan mo pa ng pagkakataon na makabawi ako sayo at muling maibalik ang tiwala mo yeah.!
Ang tiwala mo muling maibalik ang tiwala mo
(Vers)
Hindi ko na mababawi ang mga sandaling ito na sa iyo" nagkamli at ng pasakit ang puso ko"y dinudurog ng unti unti sa pagnanais na muling mag ka bati at di na muli pang sasaktan handang ibalik sayo lahat ng oras ng
Hindi mo hiningi sakin at kusang binigay ko ang lahat bagay na hiniling ko ay.agad agad na tinutupad mo sadyang magulo nga ang isipan ko nung mga panahong yun mga pangako sa isa"t isa mula nung makita kita d parin nagbabago mukan tanga kong iniisip ka kung magbabalik pa ba ako wari"y
Muli mong bubuksan ang pintuan mga panahon at oras ginto ikaw lamang ang nagpapasaya sa aking mundo ngunit wala na ang iyong tiwala saan na napadpad
Ang tiwala mo mahaaaal!
(Chorus)
Alam kong nagkamali ako nagkamali ako sayo
Di ko naman sinasadya sanay mapatawad mo ako"y sanay bigyan mo pa ng pagkakataon na makabawi ako sayo at muling maibalik ang tiwala mo yeah.!
Ang tiwala mo muling maibalik ang tiwala mo
Sanay patawarin mo na ang katulad ko oh. ohuuhh (pinag sisisihan ko lahat ng mga kasalanan ko sayo(sabihin mo sakin ano ba ang kailangan kong gawin(para lang malaman mo na ang lahat ng to ay totoo ang nais ko lang ay muling maibalik (ang tiwala mo mahaaaal! )4X ohohh
(Chorus Again)
(Vers)
Sanay mapatawad mo sanay mapatawad mo ako sa mga kasalanan kong di ko inakalang na masasaktan ko ang tulad mo ang puso mo ay nasugatan ng grabe oh grabe at ang dami ang dami mong nabalitaan sa akin di ko aminin ang saloobin ko ngaun darating ang panahon malalaman mo kung bakit ang puso ko ay sa iba na nga lumingon dahil sa naging panatag ang sarili ko dahil alam kong d mo kaya iwanan ang ka2lad ko tinitiis ang bawat sandaling parang dinadaandaanan lang kita ang utak moy gulong gulo halos d ko na mabilang ang mga kasalanan kong patuloy kong ginagawa hindi mo inaasahan na dito na pala matatapos ang relasyon natin dalawa natin na kay tagal natin iningatan patawarin mo sana ako alam kong mali ito pero pinipilit ko sana pag gising ko bukas ng umaga ay andyan kalang saking tabi ikaw parin at wala paring pinag bago(oohhh... at ang lahat ay nasa ayos na salamat sa tiwala mo